Financial Modeling with Crystal · Portfolio Models 132 Single-Period Crystal Ball Model 132...

Financial Modelingwith Crystal

Ball and Excel

JOHN CHARNES

John Wiley & Sons, Inc.


Ball and Excel

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Ball and Excel

JOHN CHARNES

John Wiley & Sons, Inc.

Copyright c© 2007 by John Charnes. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Charnes, John Martin.Financial modeling with crystal ball and Excel / John M. Charnes.

p. cm.—(Wiley finance series)‘‘Published simultaneously in Canada.’’Includes bibliographical references.ISBN 13: 978-0-471-77972-8 (paper/cd-rom)ISBN 10: 0-471-77972-5 (paper/cd-rom)

1. Finance–Mathematical models. 2. Investments–Mathematical models. 3.Crystal ball (Computer file) 4. Microsoft Excel (computer file) I. Title.

HG106.C485 2007332.0285′554—dc22

2006033467

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

In Memory of Gerald Daniel Charnes, 1925–2005

Contents

Preface xi

Acknowledgments xv

About the Author xvii

CHAPTER 1Introduction 1

Financial Modeling 2Risk Analysis 2Monte Carlo Simulation 4Risk Management 8Benefits and Limitations of Using Crystal Ball 8

CHAPTER 2Analyzing Crystal Ball Forecasts 11

Simulating A 50–50 Portfolio 11Varying the Allocations 21Presenting the Results 27

CHAPTER 3Building a Crystal Ball Model 28

Simulation Modeling Process 28Defining Crystal Ball Assumptions 29Running Crystal Ball 32Sources of Error 33Controlling Model Error 35

CHAPTER 4Selecting Crystal Ball Assumptions 36

Crystal Ball’s Basic Distributions 36Using Historical Data to Choose Distributions 54Specifying Correlations 63

vii

viii CONTENTS

CHAPTER 5Using Decision Variables 71

Defining Decision Variables 71Decision Table with One Decision Variable 73Decision Table with Two Decision Variables 79Using OptQuest 89

CHAPTER 6Selecting Run Preferences 95

Trials 95Sampling 98Speed 101Options 103Statistics 104

CHAPTER 7Net Present Value and Internal Rate of Return 105

Deterministic NPV and IRR 105Simulating NPV and IRR 107Capital Budgeting 111Customer Net Present Value 121

CHAPTER 8Modeling Financial Statements 125

Deterministic Model 125Tornado Chart and Sensitivity Analysis 126Crystal Ball Sensitivity Chart 129Conclusion 129

CHAPTER 9Portfolio Models 132

Single-Period Crystal Ball Model 132Single-Period Analytical Solution 134Multiperiod Crystal Ball Model 135

CHAPTER 10Value at Risk 140

VaR 140Shortcomings of VaR 142CVaR 142

Contents ix

CHAPTER 11Simulating Financial Time Series 147

White Noise 147Random Walk 149Autocorrelation 150Additive Random Walk with Drift 154Multiplicative Random Walk Model 157Geometric Brownian Motion Model 160Mean-Reverting Model 164

CHAPTER 12Financial Options 170

Types of Options 170Risk-Neutral Pricing and the Black-Scholes Model 171Portfolio Insurance 174American Option Pricing 177Exotic Option Pricing 179Bull Spread 183Principal-Protected Instrument 184

CHAPTER 13Real Options 187

Financial Options and Real Options 187Applications of ROA 188Black-Scholes Real Options Insights 191ROV Tool 193Summary 200

Appendix A Crystal Ball’s Probability Distributions 202

Appendix B Generating Assumption Values 235

Appendix C Variance Reduction Techniques 243

Appendix D About the Download 249

Glossary 251

References 255

Index 263

Preface

I wrote this book to help financial analysts and other interested parties learn howto build and interpret the results of Crystal Ball models for decision support.

There are several books that exist to inform readers about Monte Carlo simulationin general. Many of these general books are listed in the References section ofthis book. This book focuses on using Crystal Ball in three main areas of finance:corporate finance, investments, and derivatives.

In 1982, University of Minnesota–Duluth Business School professor HenryPerson introduced me to IFPS, computer software designed for financial planning,that we ran on VAX mainframe computers for an MBA class in quantitativemethods. IFPS used a tabular layout for financial data similar to Excel’s, although itwas more abstract than Excel’s because one had to print the data to see the layoutin IFPS instead of working with Excel’s tabular display of the data on the screen.Gray (1996) describes what is evidently the latest, and perhaps final, version of thisfinancial planning software. It is significant to me because IFPS included a MonteCarlo command that gave me my first glimpse of using a computer as a tool forfinancial risk analysis.

I was hooked. The next term, I took Henry’s class in discrete-event simulationbased on Tom Schriber’s (1974) red GPSS textbook. I found the notion of systemsimulation fascinating. It made experimentation possible in a computer lab onmodels of real-world situations, just as the physical scale models of dams in theUniversity of Minnesota–Twin Cities hydraulic laboratory made experimentationpossible for the civil engineering professors during my days as an undergraduatestudent there. I saw many places where systems simulation could have been appliedto the construction industry when I worked as a field engineer, but was unaware atthe time of what simulation could accomplish.

More graduate school beckoned. After a year of teaching finance at the Universityof Washington in Seattle, I returned to the Twin Cities to eventually earn mydoctorate in what became the Carlson School of Management. There I met DavidKelton in 1986. His coauthored textbook, now in print as Law and Kelton (2000),got me started on my dissertation research that was done largely at the MinnesotaSupercomputer Institute, where I ran FORTRAN programs on Cray supercomputersand graphed the resulting output on Sun workstations. It is amazing to me thatanyone can do the same tasks today faster and more easily by using Crystal Ball ona personal computer. I wish that I had had today’s version of the personal computerand Crystal Ball available to me when I worked as an economic analyst at a Fortune50 banking conglomerate in 1985.

xi

xii PREFACE

As assistant professor in the management sciences department at the Universityof Miami in Coral Gables, Florida, I taught simulation to systems analysis andindustrial engineering students in their undergraduate and graduate programs.When I moved to the University of Kansas in 1994, I had hopes of offering a similarcourse of study, but learned quickly that the business students here then were moreinterested in financial risk analysis than systems simulation. In 1996, I offered myfirst course in risk analysis at our suburban Kansas City campus to 30 MBA students,who loved the material but not the software we used—which was neither IFPS norCrystal Ball.

I heard many complaints that term about the ’’clunky software that crashed allthe time,’’ but one student posed an alternative. She asked if I had heard of CrystalBall, which was then in use by a couple of her associates at Sprint, the KansasCity–based telecommunications company. I checked it out, and the more I read inthe Crystal Ball documentation, the more convinced I became that the authors wereinfluenced by the same Law and Kelton text that I had studied in graduate school.

At the 1997 Winter Simulation Conference, I met Eric Wainwright, chieftechnical officer at Decisioneering, Inc. (DI), and one of the two creators of CrystalBall, who confirmed my suspicions about our shared background. Thus beganmy friendship with DI that led to creation of Risk Analysis Using Crystal Ball,the multimedia training CD-ROM offered on the DI Web site. That effort, incollaboration with Larry Goldman, Lucie Trepanier, and Dave Fredericks, was awholly enjoyable experience that gave me reason to believe—correctly—that theeffort to produce this book would also be enjoyable.

About the same time I met Eric, I had the good fortune to work with DavidKellogg at Sprint. His interest in Crystal Ball and invitation to present a series oflectures on its use as a decision support tool led to my development of trainingclasses that were part of the Sprint University of Excellence offerings for severalyears. I am grateful to David and all the participants in those classes over theyears for their helping me to hone the presentation of the ideas contained in thisbook. I am also grateful to Sprint and Nortel Networks for the financial supportthat led to development of the real options valuation tool described in Chapter 13.Other consulting clients will go unnamed here, but they also have influenced thepresentation.

Microsoft Excel has become the lingua franca of business. Business associates indifferent industries and even some in different divisions of the same company oftenfind it difficult to communicate with each other. However, virtually everyone whodoes business planning uses Excel in some capacity, if not exclusively. Though notalways able to communicate in the same language, businesspeople around the globeare able to share their Excel spreadsheets. As with everything in our society, Excelhas its critics. Yet the overwhelming number of users of this program make it foolishto deliberately shun its use.

My main criticism of Excel is obviated by use of the Crystal Ball application.Excel is extremely versatile in its ability to allow one to build deterministic models inmany different business, engineering and scientific domains. Without Crystal Ball, it

Preface xiii

is cumbersome to use Excel for stochastic modeling, but Crystal Ball’s graphical inputand output features make it easy for analysts to build stochastic models in Excel.

In the 1970s, Jerry Wagner and the other founders of IFPS had a dreamof creating software that would dominate the market for a computerized, plain-language tool for financial planning by executives. In the meantime, Microsoft Excelcame to dominate the market for financial planning software. The combination ofExcel, Crystal Ball, and OptQuest provides a powerful way for you to enhance yourdeterministic models by adding stochastic assumptions and finding optimal solutionsto complex real-world problems. Building such models will give you greater insightinto the problems you face, and may cause you to view your business in a new light.

ORGANIZATION OF THIS BOOK

This book is intended for analysts who wish to construct stochastic financial models,and anyone else interested in learning how to use Crystal Ball. Instructors witha practical bent may also find it useful as a supplement for courses in finance,management science, or industrial engineering.

The first six chapters of this book cover the features of Crystal Ball andOptQuest. Several examples are used to illustrate how these programs can be usedto enhance deterministic Excel models for stochastic financial analysis and planning.The remaining seven chapters provide more detailed examples of how Crystal Balland OptQuest can be used in financial risk analysis of investments in securities,derivatives, and real options. The technical appendices provide details about themethods used by Crystal Ball in its algorithms, and a description of some methods ofvariance reduction that can be employed to increase the precision of your simulationestimates. All of the models described in the book are available on the accompanyingCD-ROM, as is a link to a Web site from which a trial version of Crystal Ball maybe downloaded. The contents of each chapter and appendix are listed below:

Chapter 1 provides an overview of financial modeling and risk analysis throughMonte Carlo simulation. It also contains a discussion of risk management and thebenefits and limitations of Crystal Ball.

Chapter 2 describes how to specify and interpret Crystal Ball forecasts, the graph-ical and numerical summaries of the output measures generated during simulation.A retirement portfolio is used for an example.

Chapter 3 takes a helicopter view of building a Crystal Ball model. It startsout with a simple, deterministic business planning Excel model, and then showsyou how to add stochastic assumptions to it with Crystal Ball. The chapter alsocontains a discussion of possible sources of error in your models and how they canbe controlled.

Chapter 4 contains a deeper look at specifying Crystal Ball assumptions. Itdescribes Crystal Ball’s basic distributions and shows you how to select distributionsusing historical data and/or your best expert judgment. The chapter also describeshow to use, estimate, and specify correlations between assumptions in a Crystal Ballmodel.

xiv PREFACE

Chapter 5 covers the use of decision variables in detail. A decision variable isan input whose value can be chosen by a decision maker. Decision variables enableyou to harness the power of Crystal Ball and OptQuest to find optimal solutions. Afirst look at real options is included in this chapter.

Chapter 6 lists and explains the runtime options available in Crystal Ball as wellas how and when to use them.

Chapter 7 discusses the relative merits of using the concepts of net present valueand internal rate of return in deterministic and stochastic models. Examples includecapital budgeting in finance and customer lifetime value in marketing.

Chapter 8 describes how to add stochastic assumptions to pro forma financialstatements, then perform sensitivity analyses using tornado charts and Crystal Ballsensitivity charts.

Chapter 9 presents examples of using Crystall Ball to construct single- andmultiperiod portfolio models. It also compares the Crystal Ball results for a single-period model to the analytic solution in a special case where an analytic solutioncan be found.

Chapter 10 discusses Value at Risk (VaR) and its more sophisticated cousin,Conditional Value at Risk (CVaR), the relative merits of VaR and CVaR, and howthey are used in risk management.

Chapter 11 describes how to simulate financial time series with Crystal Ball. Itcovers random walks, geometric Brownian motion, and mean-reverting models, aswell as a discussion of autocorrelation and how to detect it in empirical data.

Chapter 12 shows how to create Crystal Ball models for financial option pricing,covering European, American, and exotic options. It includes a model to demonstratehow to simulate returns from option strategies, using a bull spread as an example. Italso shows how to use Crystal Ball to evaluate a relatively new derivative security, aprincipal-protected instrument.

Chapter 13 concludes the main body of the text with a discussion of how CrystalBall and OptQuest are used to value real options. It also contains a brief review ofthe literature and some applications of real options analysis.

Appendix A contains short descriptions of each available Crystal Ball assump-tion. Each description includes the assumption’s parameters, probability mass ordensity function, cumulative distribution function, mean, standard deviation, andnotes about the distribution and/or its usage.

Appendix B provides a brief description of how Crystal Ball generates therandom numbers and variates during the simulation process.

Appendix C describes some variance reduction techniques, methods by whichan analyst changes a model to get more precise estimates from a fixed number oftrials during a simulation.

Appendix D provides information on downloading the Crystal Ball softwareand Excel files that are used in this book.

Appendix E contains citations for the references in the text to academic andpractitioner literature relating to financial modeling and risk analysis. A glossary isalso included.

Acknowledgments

F or their conversations and help (unwitting, by some) in writing this bookI would like to thank: Chris Anderson, Bill Beedles, George Bittlingmayer,

David Blankinship, Eric Butz, Sarah Charnes, Barry Cobb, Tom Cowherd Jr., RizaDemirer, Amy Dougan, Bill Falloon, Dave Fredericks, Larry Goldman, DouglasHague, Emilie Herman, Steve Hillmer, Mark Hirschey, Joe B. Jones, David Kellogg,Paul Koch, Mike Krieger, Chad Lander, Michael Lisk, Howard Marmorstein,Samik Raychaudhuri, Catherine Shenoy, Prakash Shenoy, Steve Terbovich, MichaelTognetti, Lucie Trepanier, Eric Wainwright, Bruce Wallace, and Laura Walsh.Special thanks go to Suzanne Swain Charnes for help with editing and time taken toindulge my interest in Crystal Ball over the years.

I enjoyed writing this book, and hope that it helps you learn how to buildstochastic models of realistic situations important to you. I will appreciate anyfeedback that you send to [email protected].

John CharnesLawrence, Kansas 2006

xv

About the Author

D r. John Charnes is professor and Scupin Faculty Fellow in the finance, economics,and decision sciences area at the University of Kansas School of Business,

where he has received both teaching and research awards. Professor Charnes hastaught courses in risk analysis, computer simulation, statistics, operations, qualitymanagement, and finance in the business schools of the University of Miami (Florida),University of Washington (Seattle), University of Minnesota (Minneapolis), andHamline University (St. Paul).

He has published papers on financial risk analysis, statistics, and other topicsin Financial Analysts Journal, The American Statistician, Management Science,Decision Sciences, Computers and Operation Research, Journal of the OperationalResearch Society, Journal of Business Logistics, and Proceedings of the WinterSimulation Conference. Professor Charnes has performed research, consulting, andexecutive education for more than 50 corporations and other organizations inKansas, Missouri, Washington, Minnesota, Florida and Ontario, Canada.

Professor Charnes holds PhD (1989), MBA (1983), and Bachelor of CivilEngineering (1980) degrees from the University of Minnesota. Before earning hisdoctorate, he worked as a surveyor, draftsman, field engineer, and quality-controlengineer on numerous construction projects in Minnesota, Iowa, and Maryland. Hehas served as president of the Institute for Operations Research and the ManagementSciences (INFORMS) College on Simulation, and proceedings coeditor (1996) andprogram chair (2002) for the Winter Simulation Confererences.

xvii

CHAPTER 1Introduction

L ife is stochastic. Although proponents of determinism might state otherwise,anyone who works in business or finance today knows quite well that future

events are highly unpredictable. We often proceed by planning for the worst outcomewhile hoping for the best, but most of us are painfully aware from experience thatthere are many risks and uncertainties associated with any business endeavor.

Many analysts start creating financial models of risky situations with a basecase constructed by making their best guess at the most likely value for each of theimportant inputs and building a spreadsheet model to calculate the output valuesthat interest them. Then they account for uncertainty by thinking of how each inputin turn might deviate from the best guess and letting the spreadsheet calculate theconsequences for the outputs. Such a ‘‘what-if’’ analysis provides insight into thesensitivity of the outputs to one-at-a-time changes in the inputs.

Another common procedure is to calculate three scenarios: best case, worst caseand most likely. This is done by inserting the best possible, worst possible, and mostlikely values for each key input, then calculating the best-case outputs when eachinput is at its best possible value; the worst-case outputs when each input is at itsworst possible value; and using the base case as the most likely scenario. Scenarioanalysis shows the ranges of possibilities for the outputs, but gives no idea of thelikelihood of output values falling between the extremes.

What-if and scenario analysis are good ways to get started, but there are moresophisticated techniques for analyzing and managing risk and uncertainty. This bookis designed to help you use the software programs Crystal Ball and Excel to developfinancial models for risk analysis. The spreadsheet program Excel has dramaticallychanged financial analysis in the past 30 years, and Crystal Ball extends the capabilityof Excel by allowing you to add stochastic assumptions to your spreadsheets. Addingstochastic assumptions provides a clearer picture of the possibilities for each of theoutputs of interest. Reading this book and following the examples will help you useCrystal Ball to enhance your risk analysis capabilities.

Throughout the book, I use the word stochastic as a synonym for randomor probabilistic, and as an antonym for deterministic. The majority of spreadsheetmodels in use today are deterministic, but every spreadsheet user knows at some levelthat there is a degree of uncertainty about each of the inputs to his or her models.

1

2 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL

Crystal Ball enables you to use a systematic approach to account for uncertainty inyour spreadsheet models.

The first six chapters of this book demonstrate how to use Crystal Ball. Theremainder of the text provides examples of using Crystal Ball models to help solveproblems in corporate finance, investments, and financial risk management. Theappendices provide technical details about what goes on under the hood of theCrystal Ball engine.

This chapter is an overview of financial modeling and risk analysis. Someexample applications are listed below where these tools provide insights that mightnot otherwise come to light, and you get a glimpse of how straightforward it isto assess financial risk using Crystal Ball and Excel. For a simple model that isalready built and ready to run, we will interpret the output and analyze the model’ssensitivity to changes in its inputs. The chapter concludes with a discussion of thebenefits and limitations of risk analysis with Crystal Ball and Excel.

FINANCIAL MODELING

For the purposes of this book, financial modeling is the construction and useof a spreadsheet depiction of a company’s or an individual’s past, present, orfuture business operations. To learn more about deterministic financial modeling,see Proctor (2004), Sengupta (2004), or Koller, Goedhart, and Wessels (2005).For each situation where we wish to use a stochastic model, we begin with adeterministic Excel model, then add stochastic assumptions with Crystal Ball togenerate stochastic forecasts. By analyzing the stochastic forecasts statistically, wecan make inferences about the riskiness of the business operations described by themodel. The risk analysis process became much easier and more widely availablewith the introduction of Crystal Ball to the marketplace in 1987.

RISK ANALYSIS

The first recorded instances of risk analysis are the practices of the Asipu peopleof the Tigris-Euphrates valley about 3200 B.C. (Covello and Mumpower 1985).The Asipu would serve as consultants for difficult decisions such as a proposedmarriage arrangement, or the location of a suitable building site. They would listthe alternative actions under consideration and collect data on the likely outcomesof each alternative. The priest-like Asipu would interpret signs from the gods,then compare the alternatives systematically. Upon completion of their analysis,they would etch a final report to the client on a clay tablet, complete with arecommendation of the most favorable alternative (Oppenheim 1977).

According to the Oxford English Dictionary (Brown 2002), the term riskanalysis means the ‘‘systematic investigation and forecasting of risks in businessand commerce.’’ The word risk comes through French, Latin, and Italian from

Introduction 3

the Greek word rhiza, in reference to sailors navigating among cliffs. Note thatalthough some authorities believe that risk is derived from the Arabian word rizq,meaning ‘‘subsistence’’ it is difficult to explain how this meaning developed intothat of ‘‘danger’’ (Klein 1967). If you bought this book to help you analyze businessproblems, I will bet that you have no trouble seeing the connection between the risksof managing a business and the perils of navigating a sailing vessel around cliffs andbarely submerged rocks that can damage the hull and sink the ship.

Imagine an ancient Greek mariner piloting a ship as it approaches a cliff orpoint of rocks in uncharted waters. Another sailor is on lookout in the crow’s nestat the top of the mast to give the earliest possible warning about how far down intothe water an outcrop from the cliff might be. A navigator nearby with sextant andcompass is keeping track of where the ship has been and the direction in which it’sheaded. His lookout warns him at the first sign of trouble ahead, but it is up to thepilot to decide how wide to take the turn around the cliff. Cutting the corner tooclose can save time but might sink the ship. Veering far from the edge is safer, butadds costly travel time.

In navigating a strait between two cliffs, the pilot’s decision is even more difficult.Being too far from one cliff can mean being too close to the opposing cliff. The pilotmust weigh the risks, use judgment and instinct to carefully choose a course, andthen hope for the best as vessel and crew proceed through the strait.

It is the pilot’s job to take all of the available information into account and decidehow best to sail the ship in uncharted waters. The pilot wants a clear analysis of allthe dangers and opportunities that lie ahead, in order to decide whether the potentialtime savings of the ship’s chosen course outweigh the disastrous consequences ofhull damage. Even though the ship may have been through many different straitsin the past, the pilot needs a systematic investigation and forecasting of the risksassociated with the planned course through each new strait encountered during thevoyage.

If you are running a business (or are an analyst helping to run a business),you are often in situations conceptually similar to those facing the pilot of a sailingvessel in uncharted waters. You know where your business has been, and you arealways on the lookout for dangers and opportunities on the horizon. You operatein an environment fraught with uncertainty. You know that future circumstancescan affect you and your business greatly, and you want to be prepared for whatmight happen. In many situations, you need to weigh the favorable and unfavorableconsequences of some decision and then choose a course of action. Similar to aship pilot, it is your job to decide how best to navigate the straits of your businessenvironment. What do you do?

Fortunately, mathematicians such as Simon LaPlace and Blaise Pascal developedthe fundamental underpinnings of risk analysis in the seventeenth century by devisingthe mathematical methods now used in probability theory (Ore 1960). From theseprecepts came the science of statistics. ‘‘What?’’ you ask, ‘‘I studied probability andstatistics in college and hated every minute of them. I thought I was done with thatstuff. How can it help me?’’


In short, probability and statistics help you weigh the potential rewards andpunishments associated with the decisions you face. This book shows you howto use Crystal Ball to add probabilistic assumptions and statistical forecasts tospreadsheet models of a wide variety of financial problems. In the end, you still mustmake decisions based on your best judgment and instincts, but judicious use of themethods of probability and statistics that we go through in this book will help youin several ways.

The modeling process described here enables you to investigate many differentpossibilities, hone your intuition, and use state-of-the-art software tools that areextremely beneficial for managing risk in dynamic business environments. The riskanalysis process forces you to think through the possible consequences of yourdecisions. This helps you gain comfort that the course of action you select is the bestone to take based on the information available at the time you make the decision.Risk analysis is the quantification of the consequences of uncertainty in a situationof interest, and Crystal Ball is the tool for carrying it out.

MONTE CARLO SIMULATION

Risk analysis using Crystal Ball relies on developing a mathematical model in Excelthat represents a situation of interest. After you develop the deterministic model,you replace point estimates with probability distribution assumptions and forecastthe distribution of the output. The forecasted output distribution is used to assessthe riskiness of the situation.

For simple models, the output distribution can be found mathematically to givean analytic solution. For example, consider the simple cost equation

(Total Cost) = $100 + $15 × (Quantity Produced),

where (Quantity Produced) is modelled as a normal probability distribution withmean, µ = 50, and standard deviation, σ = 10, and we want to know the probabilitythat (Total Cost) is greater than $900. We don’t need Crystal Ball for this situationbecause we can easily obtain an analytic solution.

A result in probability theory holds that if a random variable X follows thenormal distribution with mean, µ, and standard deviation, σ , then the randomvariable Y = a + bX will also be normally distributed with mean, a + bµ, andstandard deviation, bσ . Therefore, we can easily determine that (Total Cost) isnormally distributed with mean, 100 + (15 × 50) = $850, and standard deviation,15 × 10 = $150. Using a table of cumulative probabilities for the standard normaldistribution, or using the Excel function = 1-NORMDIST (900,850,150, TRUE), wecan find that the probability is 36.94 percent that (Total Cost) is greater than $900.See the file Analytic.xls for these calculations, along with a Crystal Ball model thatvalidates the solution.

Introduction 5

FIGURE 1.1 Simple profit model inProfit.xls. Cells C5:C7 are defined asCrystal Ball assumptions. Cell C11 is aCrystal Ball forecast.

In practice however, it is easy to find situations that are too difficult for mostanalysts to solve analytically. For example, consider a simple situation where unitsales, S, follow the Poisson distribution with mean 10; price, P, is lognormallydistributed with mean $50 and standard deviation $10; variable cost percentage,V, has the beta distribution with parameters minimum = 0%, maximum = 100%,alpha = 2, and beta = 3; and fixed cost, F = $100. Then profit, π , is calculated as

π = SP(1 − V) − F (1.1)

in the Profit.xls model shown in Figure 1.1. The stochastic assumptions in Profit.xlsare shown in Figure 1.2. For this model, it is easy to obtain a forecast distributionfor profit with Crystal Ball and find the probability of making a positive profit to beapproximately 76 percent (Figure 1.3), but it is not so easy to determine the distribu-tion mathematically and obtain the probability of positive profit by analytic solution.

Further, Crystal Ball enables us do sensitivity analysis very easily. The sensitivitychart in Figure 1.4 shows that most of the variation in profit arises from theAssumption V, the variable cost percentage. If we are able to control this variable,then the model shows that we can reduce the variation of profit by reducing thevariation in V. If this variable is beyond our control, the chart indicates that weshould work first on V when we are fine-tuning the model by improving ourestimates of the assumption variables.

With Crystal Ball, we obtain an approximate solution using Monte Carlosimulation to generate the output distribution. One of the features of Monte Carlo


FIGURE 1.2 Crystal Ball assumptions defined in cells C5:C7 of the simpleprofit model.

is that the more simulation trials we run, the closer is our approximation to thetrue distribution. The technique of Monte Carlo simulation has been used for thispurpose for many years by scientists and engineers working with large and expensivemainframe computers. In combination with today’s small and inexpensive personalcomputers, Crystal Ball and Excel bring to everyone the ability to run Monte Carlosimulations on a PC.

In statistics, one of the earliest uses of simulation (albeit without a computer),was that by the mathematical statistician William S. Gossett, who used the penname ‘‘Student’’ to conceal his identity and appease his employers at Guinnessbrewing company. To verify his mathematical derivations, Gossett repeatedly drewrandom samples of numbers from a bowl, wrote them down, and painstakinglymade his calculations. The results he obtained were within the tolerances expectedfrom an experiment involving random sampling. Gossett’s sampling experiments areconceptually similar to what we do on computers today, but Crystal Ball does for usin seconds what must have taken Gossett weeks or months to do by hand in his day.

Introduction 7

FIGURE 1.3 Forecast distribution for the simple profit model. Thecertainty of 75.74 percent for the range from 0.00 to infinity is the CrystalBall estimate of the probability of positive profit, Pr(π > 0) for the modelspecified by Expression 1.1.

FIGURE 1.4 Sensitivity chart for the simple profit model. The inputvariables (assumptions) P, S, and V are listed from top to bottom indecreasing order of their impact on the output variable (forecast), profit.


The term Monte Carlo originated in a conversation between two mathematiciansemployed by Los Alamos National Laboratory as a code word for their secret workon the atomic bomb (Macrae 1992). John von Neumann and Stanislaw Ulamapplied Monte Carlo methods to problems involving direct simulation of behaviorconcerned with random neutron diffusion in fissionable material (Rubinstein 1981).The name was motivated by the similarity of the computer-generated results to theaction of the gambling devices used at the casinos in the city of Monte Carlo inthe principality of Monaco. The term caught on and is now widely used in finance,science, and engineering.

RISK MANAGEMENT

When analyzing risk with the methods presented in this book you will be able toquantify the consequences of uncertainty by answering three main questions:

1. What can happen?2. How likely is it to happen?3. Given that it occurs, what are the consequences?

All good managers go through a process by which they consider these questionssomehow, even if subconsciously. By taking the time to answer the questions inquantifiable terms, you will develop deeper insight into the problems you face.

Risk analysis is part of a broader set of methods called risk management, whichalso seeks to find answers to three main questions:

1. What can be done?2. What options are available?3. What are the associated trade-offs in terms of costs, benefits, and risks?

It is your job as a manager or analyst to identify what can be done and what optionsare available, but once you have done so, the methods in this book will help youinvestigate the associated trade-offs in terms of costs, benefits, and risks.

Risk analysts often claim that much of the benefit of using Crystal Ball comesjust from going through the process used to develop and fine-tune their models. Therisk analysis process helps you develop insights into a problem more quickly thanyou would without it. In a sense, the intent of this book is to help save you some ofthe time and tuition you might otherwise pay in the school of hard knocks to reachgood decisions.

BENEFITS AND LIMITATIONS OF USING CRYSTAL BALL

This section describes some of the benefits and the limitations of using Crystal Balland Excel for risk analysis. When it is applicable, diligent use of these tools yields

Introduction 9

deeper insight and understanding that will lead to better decision making. However,the tools have their limits as described below.

Benefits■ Careful study of the situation being modeled usually reveals the key input factors

that lead to success. Sometimes it will become obvious what these factors areduring model building. However, Crystal Ball has built-in sensitivity analysistools to help identify the key input factors.

■ As mistakes are costly, it is better to evaluate before implementation. A validmodel of a situation can help save much time and expense compared toexperimenting with aspects of the actual situation to see what happens. Thisbook is intended to help you build valid models.

■ Computer hardware and software make simulation easy. Until recently, theuse of Monte Carlo simulation was limited to those who had access to largemainframe computers and the expertise to program them. Today’s personalcomputers have the same computational horsepower as yesterday’s mainframes.Crystal Ball has been developed over the years to make the addition of stochasticinputs and the calculation of output statistics as easy as possible.

■ Realistic situations can be analyzed with relatively simple models. While manyrealistic situations have a host of potential complications, most often just a fewof these have the greatest effect on the outputs. Crystal Ball’s sensitivity analysisfeatures help you identify the effects of the factors having the greatest impact.

■ Risk analysis can be a convincing agent for change. For those who understandthe modeling process and the output produced by Crystal Ball, experimentingwith the model can demonstrate very well the impact of changes to the system.If you have earned the buy-in of key decision makers, the model can be apowerfully persuasive tool.

Limitations■ Validity of the input data is essential. As with any other computer program, the

output is only as good as the input. The aphorism ‘‘garbage in, garbage out’’will always be true for Crystal Ball models as it will be for every computerprogram.

■ If you can’t model it, you can’t simulate it. Your ability to build Crystal Ballmodels is subject to the limitations of Excel. If you cannot build an Excelspreadsheet to represent the situation, then you cannot use Crystal Ball forsimulation. Fortunately, the hundreds of functions built into Excel make it veryversatile, and Crystal Ball allows you to use Excel’s VBA capabilities if necessaryfor specialized purposes.

■ Risk analysis requires expert insight to make decisions. Crystal Ball will notmake decisions for you. It will help you gain insight into the problem, but youmust still reach conclusions and make decisions based on your judgment andintuition about the situation you are analyzing.


■ Crystal Ball gives approximate rather than exact solutions. This ‘‘limitation’’ ismostly an academic criticism. Many purists prefer to use only models that availthemselves to analytic solution. When confronted by a realistic complication thatprecludes analytic solution, they simply assume away the complication. Mostpractitioners, however, wish to include realistic complications and are happyto accept the trade-off of getting an approximate solution with simulation. Thelimitation is that they might have to run the model longer to obtain the desiredprecision. However, most practitioners prefer an approximate solution to arealistic problem over an exact solution to an oversimplified problem that isonly a rough approximation to reality.

CHAPTER 2Analyzing Crystal Ball Forecasts

I n this chapter, using an example of accumulating funds for retirement, we seethe graphical and numerical summaries of forecasts that Crystal Ball provides

automatically. This chapter serves as a review of elementary statistical analysis,focused on the standard output built into Crystal Ball.

SIMULATING A 50 – 50 PORTFOLIO

Let’s say that you want to start saving for your retirement. You are 30 years old andwish to retire at age 60. You plan to put away an inflation-adjusted $10,000 peryear, and would like to know how much wealth you will have accumulated after 30years. At this point, you consider only two types of assets: stocks and bonds.

If you had perfect foresight, you would know exactly what returns eachinvestment would bring over the next 30 years. With that information, you wouldn’tneed Crystal Ball and could optimize your portfolio by investing in only those assetsthat you knew would go up. Of course, no one has perfect foresight, so what do youdo? In this chapter, we’ll consider an oversimplified model for investing retirementfunds and use it to illustrate how to analyze Crystal Ball forecasts.

Accumulate.xls

Overview. For this model we assume that returns on stocks and bonds duringthe next 30 years will resemble (in a statistical sense) the returns thathave been observed during the years 1926 to 2004. You have heard thatdiversification is a good thing to do when investing your money, so youdecide initially to split the money you want to set aside for retirement eachyear by putting one half into stocks and one half into bonds. We call thisthe 50–50 portfolio, and model it in the Excel file Accumulate.xls.

The model draws returns on stocks and bonds randomly for each year andcalculates the wealth you will have accumulated for retirement 30 yearsfrom now, assuming a constant 3 percent inflation rate. A segment of thismodel is shown in Figure 2.1. Note that rows 14 through 40 are hidden tosave space.

11


FIGURE 2.1 Spreadsheet segment from Accumulate.xls showing accumulated value after 30 years ofretirement savings where 50 percent was put into stocks and 50 percent into bonds each year. Note thatRows 14 through 40 are hidden.

Forecast. You want to see the possible distribution of wealth you will haveaccumulated after 30 years—Year 30 Wealth. Cell D4 has the formula=H41 to get Year 30 Wealth displayed near the top of the spreadsheet.

Stochastic assumptions. Cells B12:B41 represent the total annual return onstocks for years 1 through 30. Each year’s return on stocks is lognormalwith mean 1.1517 and standard deviation 0.2746. Cells C12:C41 representthe total annual return on bonds for years 1 through 30. Each year’s returnon bonds is lognormal with mean 1.0585 and standard deviation 0.0735. Inthis model, all assumptions are statistically independent of each other (thatis, all correlations are zero). The parameters of the lognormal distributionswere chosen with the help of Crystal Ball’s distribution-fitting feature, whichis described in Chapter 4.

Decision variable. The sole decision variable in this model is Cell B4, the pro-portion of each year’s investment that is allocated to stocks. The proportionallocated to bonds is calculated in Cell B5 as =1–B4 to ensure that theentire investment is put into either stocks or bonds each year. The initialproportion allocated to stocks is 50 percent.

Summary. Figure 2.2 shows the frequency view of a forecast chart for Year 30Wealth obtained after running the simulation for 10,000 trials. Note thatthe distribution is highly skewed to the right because of a few instances ofvery high returns over the 30-year period.

Frequency Chart

The default graph in the Forecast window is a frequency chart (also known as ahistogram) that shows how often a forecast cell had a value falling in each of severalpossible intervals. In the forecast window for Year 30 Wealth in Figure 2.2, thepossible values range from $0 to $15 million, and are broken up into 15 equal

Analyzing Crystal Ball Forecasts 13

FIGURE 2.2 Frequency chart showing accumulated value after 30 yearsof retirement savings where 50 percent was put into stocks and 50 percentinto bonds each year. The estimated probability of accumulating between$1 million and $2 million is shown in the Certainty field.

intervals (of size $1 million). The height of each bar indicates how many simulationtrials resulted in a Year 30 Wealth value that fell in the corresponding interval.

For example, the tallest bar in Figure 2.2 indicates that Year 30 Wealth wasbetween $1 million and $2 million in just fewer than 3,600 trials. The right side ofthe histogram has the frequency scale. Because there were 10,000 trials run, and theestimated probability (certainty) of Year 30 Wealth being between $1 million and$2 million is 34.47 percent, we know that there must have been 3,447 trials thatresulted in Year 30 Wealth between $1 million and $2 million. The left side of thehistogram has the probability scale. The text ‘‘9,761 Displayed’’ in the upper rightcorner of the window indicates that Year 30 Wealth was greater than $15 million(literally, ‘‘off the chart’’) on 239 of the 10,000 simulation trials run. We knowthat all of the undisplayed values must be above $15 million rather than below $0because the portfolio value can never be negative under the assumptions we used.

Cumulative Frequency Chart

Change the forecast window by clicking on Preferences, Forecast. . ., then selectingCumulative Frequency in the View field and clicking OK. You may also click onView → Cumulative Frequency in the top menu of the forecast window. The cumu-lative frequency chart in Figure 2.3 shows the frequency with which the simulationtrials fell into each interval or below. For example, from the cumulative frequency


FIGURE 2.3 Cumulative frequency chart showing accumulated valueafter thirty years of retirement savings where 50% was put into stocksand 50% into bonds each year. The estimated probability ofaccumulating more than $1 million is shown in the Certainty field.

chart you can see that in 80.09 percent of the trials, 30-year wealth had a value below$5 million. Another way to interpret this example is to say that the estimated prob-ability is approximately 80 percent that 30-year wealth will be less than $5 million.

Note that Crystal Ball also provides a reverse cumulative frequency chart (notpictured here) that shows the frequency with which the simulation trials fell intoeach interval or above. You can cycle through the different views by using thekeystroke combination Ctrl-d when the forecast window is active. Table 2.1 liststhe keystroke combinations (sometimes called ‘‘hot keys’’ or ‘‘keyboard shortcuts’’)that are available to alter your view of the forecast window. Make the Year 30Wealth forecast window active and experiment with different combinations to seethe effects.

Statistics View

The statistics view of the Forecast window in Figure 2.4 provides numerical sum-maries of the forecast. Descriptions of the statistics computed automatically byCrystal Ball in Figure 2.4 are listed below. The number of each statistic in thelist can be used in the Excel formula =CB.GetForeStatFn(Range,Index) as thevalue of Index. The statistics and their indices are also listed in Table 4.1, andusage of the =CB.GetForeStatFn(Range,Index) is illustrated in Cells D46:E57 inAccumulate.xls.


TABLE 2.1 Keystroke combinations (‘‘hot keys’’) that can be used to cycle through settingsavailable in the Chart Preferences dialog. These commands work on the primarydistribution—the theoretical probability distribution for assumptions, and the generatedvalues for forecasts and overlay charts.

Hot Key Description

Ctrl-d Cycles through chart views: Frequency, Cumulative Frequency, Reverse Cumu-lative Frequency (for assumption and forecast charts)

Ctrl-bor Ctrl-g

Cycles through bins or group interval values to adjust the number of data binsused to create the chart

Ctrl-l Cycles through gridline settings: None, Horizontal, Vertical, BothCtrl-t Cycles through chart types: Area, Line, Column; for sensitivity charts: Bar

(direction), Bar (magnitude), Pie (in Contribution to Variance view)Ctrl-3 Cycles between two-dimensional and three-dimensional chart displayCtrl-m Cycles through central tendency marker lines: None, Base Case, Mean, Median,

Mode (except for sensitivity and trend charts)Ctrl-n Toggles the legend display on and offCtrl-p Cycles through percentile markers: None, 10%, 20%, . . ., 90%Spacebar Cycles through window views when Excel is not in Edit mode: Chart, Statistics,

Percentiles, Goodness of Fit (if distribution fitting is selected—except for trendcharts)

For the mathematical expressions in the list below, we denote the valuesproduced by the simulation in a forecast cell as y1, y2, . . . , yn, where n is the numberof iterations run before the simulation stops.

1. Trials. The first item listed in the statistics view is Trials, which is also calledthe number of iterations, n. A trial (or iteration) is a three-step process in whichCrystal Ball generates a random number for each assumption cell, recalculatesthe spreadsheet model(s), and collects the result(s) for the forecast window(s).The number of trials is the only descriptive statistic value that is under yourdirect control. Use the Run Preferences dialog box to specify the maximumnumber of trials.

2. Mean. The next item in the Statistics View is Mean, which is the same as thearithmetic average. It is calculated as

Mean = Y = 1n

n∑

i=1

yi.

Even though the window indicates that only 9,761 trials are displayed, the value$3,775,824 for the mean in Figure 2.4 is calculated from all 10,000 trials of thesimulation.


FIGURE 2.4 Forecast chart statistics for accumulated value after 30 yearsof retirement savings where 50 percent was put into stocks and 50 percentinto bonds each year.

If you wish to calculate statistics for selected values of the forecast, click onPreferences, Forecast..., then the Filter tab. You will be able to include orexclude any specified range of values from the calculations. This feature willcome in handy later for calculating expected tail loss.The mean is one of three measures of location computed by Crystal Ball. Theother two measures of location are the median and the mode. In a highly skeweddistribution such as Year 30 Wealth, the mean may not be the best indication ofthe location of the distribution. Use the certainty range in the forecast chart tosee for yourself that the probability of Year 30 Wealth equalling or exceedingthe mean is only about 30 percent.

3. Median. The median is the value in the middle of the distribution. For example,6 is the median of the distribution of values 1, 3, 6, 7, and 9. In a distributionwith an odd number of values, the median is found by ordering the values fromsmallest to largest and then selecting the middle value. In a distribution withan even number of values, the median is equal to the mean of the two middleordered values.For Year 30 Wealth, the median value in Figure 2.4 is $2,425,951, which meansthat the probability is about 50 percent that our retirement portfolio will bethat large or larger at that time. The median is less sensitive to outliers than themean. For that reason, it is sometimes preferred to the mean as a measure oflocation, as it is here because the Year 30 Wealth forecast frequency distributionis highly skewed.


4. Mode. The mode is the single value that occurs most frequently in a set ofvalues. For a forecast that can take on continuous values, it is likely that nosingle value will occur more than once, so the mode is often listed as dashes(– –) in the statistics view, as it is in Figure 2.4.

5. Standard Deviation. The standard deviation is a measure of dispersion, orspread, of a distribution. Think of it as roughly equal to the average distance ofeach value from the mean, although as you can see in the formula below, it isnot exactly equal to that:

Standard Deviation = s =√√√√ 1

n − 1

n∑

i=1

(yi − y)2. (2.1)

The standard deviation is one of two equivalent measures of spread computed byCrystal Ball. The other measure of spread is the variance. In many applications,the standard deviation is preferred because it is measured in the same units asthe forecast variable.

6. Variance. The variance is another measure of dispersion that is equivalent tothe standard deviation. Because the variance is equal to the standard deviationsquared, it sometimes appears in the statistics view as a very large number. Thevariance is calculated as:

Variance = s2 = 1n − 1

n∑

i=1

(yi − y)2.

7. Skewness. Skewness is a measure of asymmetry of a frequency distribution.The distributions pictured in Figure 2.5 display negative, positive, and near-zeroskewness. The formula for skewness used by Crystal Ball is

Skewness = 1n

n∑

i=1

(yi − y

s

)3

.

The large positive skewness of Year 30 Wealth can be seen in Figure 2.2, and ismeasured as 32.20 in Figure 2.4. The fact that the mean of $3.776 million is somuch larger than the median of $2.426 million is also evidence of large positiveskewness. Alternative measures of skewness are

3(Mean − Mode)Standard Deviation

and3(Mean − Median)Standard Deviation

,

either of which you can calculate easily from the Crystal Ball output.With all else equal, positive skewness in accumulated wealth is desirable;however, note that here the large positive skewness makes it misleading toexpect to earn the mean wealth. As calculated above, there is less than a 30percent chance that you will actually accumulate the mean wealth or more.


FIGURE 2.5 Frequency distributions depicting negative (Skewness = −2), positive (+2), and near-zero(0.02) skewness at top left, top right, and bottom center, respectively.

8. Kurtosis. Kurtosis is a measure of peakedness, which is equivalent to measuringtail thickness. The formula for kurtosis used by Crystal Ball is

Kurtosis = 1n

n∑

i=1

(yi − y

S

)4

.

If two distributions have the same standard deviation, the one with the higherkurtosis will have a higher peak and heavier tails. All normal distributionshave a kurtosis of 3.0, and frequency distributions with a kurtosis near 3.0 arecalled mesokurtic. Distributions that have significantly higher kurtosis are calledleptokurtic and those with significantly lower kurtosis are called platykurtic.Some computer programs and authors subtract 3.0 from the definition aboveused by Crystal Ball. The technical name for this is then excess kurtosis, butyou may see it too called kurtosis by some authors, so beware. In fact, thestatistic calculated by Excel’s =KURT(ref) command and its Data Analysis tool isactually excess kurtosis—calculated by a slightly different formula than CrystalBall—but is labelled simply as kurtosis.Figure 2.6 shows an overlay chart for two distributions that have mean zeroand standard deviation one. The distributions with the lower peak is thestandard normal distribution, which has kurtosis = 3.0, as does every normaldistribution. The distribution labeled Mixture has kurtosis = 12.1. It could havecome from a market in which 5% of the time returns have high variability and95 percent of the time they have low variability (see the file Kurtosis.xls fordetails).


FIGURE 2.6 Overlay chart depicting a mesokurtic standard normal distribution (kurtosis = 3) and aleptokurtic mixture distribution (kurtosis = 11.7). See the file Kurtosis.xls for details.

9. Coefficient of Variability. The coefficient of variability, also known as the coef-ficient of variation, is a relative measure of dispersion found as

Coefficient of Variability = sy.

It might be more useful than standard deviation for some purposes. For example,a standard deviation of 10 may be insignificant if the mean is 10,000 (givinga coefficient of variability = .001) but may be substantial if the mean is 100(coefficient of variability = .1). Because the mean y and standard deviation shave the same units, the coefficient of variability is dimensionless.Crystal Ball calculates it routinely as part of the standard output, but the coeffi-cient of variability is best used only when all simulated values are positive. Whenthe values can be both positive or negative, the mean can be zero or negative.When the mean value is near zero, the coefficient of variability is sensitive tosmall changes in the standard deviation, which can limit its usefulness. Whenthe mean value is negative, use the absolute value of the coefficient of variabilityto get a more meaningful relative measure of dispersion.

10. Minimum. The minimum is the smallest value of all the observed forecast values.Note for models using unbounded-on-the-left stochastic assumptions such asthe normal distribution, the more trials that are run, the smaller the minimumis likely to be simply because there are more opportunities for Crystal Ball togenerate extreme observations.


FIGURE 2.7 Plot of 100/√

n for n in the interval [100, 10000]. This plotshows how the standard error of the mean decreases as a function of thenumber of trials in the simulation. Much of the decrease in standard error isgained after only 2,000 trials.

11. Maximum. The maximum is the largest value of all the observed forecast values.For assumptions that are unbounded on the right such as the normal or lognor-mal distributions, the more trials that are run, the larger the maximum is likelyto be simply because there are more opportunities for Crystal Ball to generateextreme observations.

12. The Range is the difference between the minimum and the maximum. Inversions of Crystal Ball before 7.2, the Range was listed in the statis-tics view of the forecast window. For backward compatibility, the com-mand =CB.GetForeStatFN(range,12) will return the Range statistic, whichis simply the Minimum subtracted from the Maximum. Do not confuse theCB.GetForeStatFN argument range, which represents the address of a forecastcell, with the Range statistic described here.

13. Mean Standard Error. The mean standard error is a measure of precision ofthe estimate of the mean. The smaller the mean standard error, the greaterthe precision. Figure 2.7 shows how the mean standard error decreases in anonlinear manner as the number of trials increases for a forecast distributionhaving a standard deviation of 100. Most of the precision in the estimate isgained by 2,000 trials.

Forecast Window Percentiles View

The percentiles view in Figure 2.8 gives the forecast values that are just larger thanthe corresponding percent of all the values. For example, the 80th percentile forYear 30 Wealth is $4,985,234. This means that 80 percent of the forecast valueswere less than that amount, while 20 percent were greater than that amount. Note


FIGURE 2.8 Forecast chart percentiles for accumulated value after 30years of retirement savings where 50 percent was put into stocks and 50percent into bonds each year.

that the 50th percentile of $2,425,951 is the same as the value of the median inFigure 2.4.

VARYING THE ALLOCATIONS

We started with a simple example of allocating 50 percent to each type of fund,which we call the 50–50 portfolio. Now we will compare forecasts for differentallocations of stocks and bonds to the portfolio.

Decision Table Tool

In file Accumulate.xls, select cell B4, then click on Define > Define Decision. . .. Fillin the fields in the Define Decision Variable dialog as shown in Figure 2.9. ClickOK and watch cell B4 turn yellow to indicate that a Crystal Ball decision variable isdefined there. After this definition, you can use the Decision Table tool to vary theallocation into stocks from 10% to 90% in steps of 10%.

Click on Run > Tools > Decision Table. . .. In the Specify target (step 1 of 3)dialog, Year 30 Wealth will be the only forecast listed (Figure 2.10), so click Next >.In the Select one or two decisions (step 2 of 3) dialog, Stocks will be the onlyavailable decision variable. Move this rightward to the Chosen Decision Variableswindow by clicking >> (Figure 2.11), then click Next >.


FIGURE 2.9 Dialog window showing settings to specify nine differentallocations into stocks each year. The allocation to stocks varies from 10percent to 90 percent in steps of 10 percent.

FIGURE 2.10 Step 1 in using the Decision Table tool.


FIGURE 2.11 Step 2 in using the decision table tool.

The Specify options (step 3 of 3) dialog should look like Figure 2.12. Specify2000 trials as shown in Figure 2.12 and click Start. Crystal Ball will run 2,000 trialsfor each of the nine allocations.

When Crystal Ball finishes its decision table analysis, you should see a newworksheet with nine Crystal Ball forecasts in cells B2:J2. The value in each cellis the mean of the forecast for Year 30 Wealth for the corresponding allocation tostocks that appears in Row 1 of the worksheet.

Trend Chart

Select cells B2:J2, then click Trend Chart. Your trend chart should look like thatdepicted in Figure 2.13, which clearly shows how the risk and expected rewardincreases as you increase your allocation to stocks.

The certainty bands in Figure 2.13 are centered on the median, so for each ofthe nine allocations listed on the horizontal axis, the 10 percent band extends fromthe 45th to 55th percentile, the 25 percent band extends from the 37.5th percentileto the 62.5th percentile, the 50 percent band extends from the 25th percentile tothe 75th percentile, and the 90 percent band extends from the 5th to the 95th


FIGURE 2.12 Step 3 in using the Decision Table tool.

FIGURE 2.13 Trend chart comparing accumulated values after 30 years ofretirement savings for nine different allocations into stocks and bonds each year.Year 30 Wealth (1) represents a 10 percent allocation to stocks—the 10–90portfolio, Year 30 Wealth (2) represents a 20 percent allocation to stocks—the20–80 portfolio, . . . , and Year 30 Wealth (9) represents a 90 percent allocation tostocks—the 90–10 portfolio.


FIGURE 2.14 Overlay chart comparing accumulated value after 30 years of retirement savings for twodifferent allocations into stocks and bonds each year. Year 30 Wealth (5) represents a 50 percentallocation to stocks and 50 percent allocation to bonds, and Year 30 Wealth (9) represents a 90 percentallocation to stocks and 10 percent allocation to bonds.

percentile of each forecast distribution. The slopes of the lines connecting each ofthese percentiles provide insight into the behavior of the wealth distribution as wevary the portfolio allocations to stocks and bonds.

The facts that the spread of the lines increases and that seven of the eight lines(all but the 5th percentile line) are sloped upwards indicate that greater allocation tostocks provides greater upside potential for Year 30 Wealth. The negative slope ofthe 5th percentile line indicates slightly greater downside risk as more of the portfoliois allocated to stocks, but for many investors this downside risk is outweighed bythe much greater upside potential.

Overlay Chart

In the worksheet created by Crystal Ball’s decision table analysis, select cell F2,then hold down the Ctrl Key and click on cell J2. Next, click on Overlay Chart.Use the hot keys Ctrl-t and Ctrl-d to make your chart look like that in Figure 2.14,which compares the risk profiles of a 50–50 portfolio to a 90–10 portfolio. Therisk profiles are line graphs depicting the probability on the vertical axis that eachportfolio yields a Year 30 Wealth greater than the corresponding monetary value onthe horizontal axis.


FIGURE 2.15 Overlay chart comparing accumulated value after 30 years of retirement savings for twodifferent allocations into stocks and bonds each year. Year 30 Wealth (1) represents a 10 percentallocation to stocks and 90 percent allocation to bonds, and Year 30 Wealth (9) represents a 90 percentallocation to stocks and 10 percent allocation to bonds.

The 90–10 portfolio almost completely dominates the 50–50 portfolio in thesense that the line representing the 90–10 portfolio is above and to the right of the50–50 line almost everywhere. For Year 30 Wealth values below about $1 million,the lines are virtually indistinguishable. While very risk-averse investors might preferthe 50–50 portfolio because it dominates the 90–10 portfolio slightly in the worst10 percent of the cases, most investors would prefer the 90–10 portfolio’s wealthdistribution because of its near equivalence in the lowest 10 percent and dominancein the upper 90 percent of the potential returns.

Figure 2.15 presents information comparing the 90–10 portfolio to the 10–90portfolio that is similar to that provided by Figure 2.14 for comparing the 50–50portfolio to the 90–10 portfolio. However, because Figure 2.15 is a cumulativefrequency chart, the near dominance of the 90–10 portfolio is manifested by the90–10 line lying to the right and below the line for the 10–90 portfolio for Year 30Wealth values above $1 million.


PRESENTING THE RESULTS

The example in this chapter illustrates both positive and negative aspects of simula-tion analysis.

On the positive side, Crystal Ball provides a clear picture of the entire distributionof results with views that can be easily selected to suit the tastes of your audience. Forexample, some people prefer frequency distributions, while others prefer cumulativeor reverse cumulative distributions. The statistics window automatically displaysmost of the descriptive statistics that decision makers might be interested in seeing.The percentiles window and trend chart present views that can also be helpful forinterpreting and comparing output distributions. This lets decision makers compareforecasts on a wide variety of dimensions.

A negative aspect is that output distributions can be compared on so manydimensions. Until you become accustomed to thinking about distributions of resultsrather than single measures such as the mean or median, trying to interpret theoutput can be somewhat overwhelming.

Crystal Ball gives you the ability to present the output in a wide variety of ways.Selecting the best way to communicate your results is a skill that requires some efforton your part to acquire. This book provides many examples of using Crystal Ball asa decision tool. Working through the examples and generating the output on yourcomputer as you read will help you decide what is best for you and your clients.

CHAPTER 3Building a Crystal Ball Model

M onte Carlo simulation is a tool for modeling uncertainty. Typically, we beginwith a deterministic spreadsheet model of the situation we are analyzing, then

use Crystal Ball to add stochastic assumptions to represent the most importantsources of uncertainty. In the past, stochastic modeling was an endeavor bestundertaken only by highly trained scientists and engineers working on mainframecomputers. Crystal Ball has been developed over the years to make Monte Carlosimulation accessible to financial analysts and others using Excel on personalcomputer workstations rather than coding in a programming language such as C++or FORTRAN. This chapter goes through the process of starting with a simplefinancial model and adding stochastic assumptions to a deterministic model in orderto illustrate the basics of building Crystal Ball models.

SIMULATION MODELING PROCESS

Most analysts facing a business problem follow the typical process for stochasticmodeling:

1. Develop a model that ‘‘behaves like’’ the real problem, with a special consider-ation of the assumptions—the random or probabilistic input variables.

2. Run a set of trials to learn about the behavior of the simulation model.3. Continually modify the model until it has credibility with the decision makers.4. Analyze the forecast (output) statistics and graphics to help make decisions

about the real problem.

The analysis and decisions in step 4 often will lead you to think about newproblems or variants of the old problem, in which case you should go to step 1to develop a new model or modify the old model to reflect the variant of the oldproblem.

Example: AKGolf.xls

For the upcoming golf season, Alaskan Golf (AG) wishes to order a shipment ofgolf clubs to offer for sale in its retail stores and affiliated pro shops. It plans to

28

Building a Crystal Ball Model 29

FIGURE 3.1 Deterministic spreadsheet model forAlaskan Golf purchase decision.

purchase 1,000 HeMan Stick drivers—a helium-filled club that the manufacturerpromises will elevate any player’s game. AG can purchase the drivers at a cost of$300 each, and plans to sell them during the golf season for $395 each. At the endof the season, all remaining HeMan Sticks will be sold for $195 to clear the shelves.AG has asked you to compute their profit if the demand for HeMan Sticks is 800units during the season. The file AKGolf.xls shown in Figure 3.1 has a deterministicspreadsheet model for this situation. Cell C11 shows that the profit will be $55,000from purchasing 1,000 HeMan Sticks at the beginning of the golf season, selling 800at $395 each during the season, and selling the remaining 200 at $195 each duringthe end-of-year sale to clear the shelves.

This spreadsheet can be used to do a ‘‘what-if’’ analysis, where the demandfor drivers in C6 is changed to various values to see the effects on profit in C11.This is informative but cumbersome, as the analyst must keep track of the inputvalues for C6 and the corresponding output values in C11, then somehow analyzethe results for a rudimentary form of sensitivity analysis. Monte Carlo simulationwith Crystal Ball can be regarded as a sophisticated form of ‘‘what-if’’ analysis, asthe program will keep track of all these changes during its run. Further, with CrystalBall it is easy to change more than one input at a time, as we will see in the nextsection.

DEFINING CRYSTAL BALL ASSUMPTIONS

A stochastic input cell in Crystal Ball is called an assumption, while an output cellis called a forecast. In this section, we will add assumptions and a forecast to thedeterministic model in AKGolf.xls to create a stochastic model.


TABLE 3.1 Stochastic assumptions to be added to AKGolf.xls.

Assumption Cell Distribution

Demand during Golf Season C6 Triangular(500,800,1500)Initial Price C8 Uniform(355,395)Sale Price C10 Uniform(175,195)

Defining Assumptions

Now let’s take into account two sources of uncertainty: (1) the demand for driversis stochastic, and (2) the retail clerks and golf professionals who sell the clubs atretail are allowed to give consumers slight discounts on the listed prices.

For the demand distribution, we assume that the lowest possible demand duringthe season is 500 drivers, the largest possible demand is 1,500 drivers, and the mostlikely demand is 800 drivers. These three parameters are used to define a triangularprobability distribution in the manner shown below. To keep things simple, we willassume that all drivers left unsold at the end of the season will be sold during theend-of-year sale at the sale price in Cell C10.

The golf professionals are allowed to give their customers discounts of up to 10percent on both the initial prices and the sales prices, which reduce their commissionsproportionately. To model this discounting, we assume that the average initial priceof drivers sold during the season is uniform with a minimum of $355, and a maximumof $395. The average end-of-season sales price is uniform with a minimum of $175,and a maximum of $195. Table 3.1 summarizes the Crystal Ball assumptions to beadded to the model in file AKGolf.xls.

Follow the steps below to define the assumptions in Cell C6:

1. Select cell C6, then click on Cell→Define Assumption.2. Choose the triangular distribution from the distribution gallery (Figure 3.2).3. Enter the parameter values andaname for the assumption.Bydefault,CBwill look

first for an Excel-defined named, then in the cells at immediate left and just abovefor an assumption name. Figure 3.3 shows where to enter the parameter values.

4. Click Enter to see a depiction of the defined distribution. Your distributionshould look like that shown in Figure 3.3. Press OK to return to the spreadsheet.It is OK to press OK without seeing the depiction, but pressing Enter first isgood practice to check that the entered distribution is what you intend to use.

Cell C6 will turn green to indicate that an assumption cell is defined there. Byfollowing the steps above for cell C6, it should be straightforward for you to definethe stochastic assumptions listed for cells C8 and C10 in Table 3.1.

If an error occurs in step 1, try entering an arbitrary numerical value in the cellfirst—Crystal Ball needs to recognize the location as a number cell. In Crystal Ball 7.2and higher, when you attempt to define an assumption in an empty cell, a value of 0


FIGURE 3.2 Distribution gallery showing the triangular andnormal distributions. To see all of Crystal Ball’s distributions, clickon All at upper left and use the upper set of scroll bars to movethrough the distributions.

FIGURE 3.3 Defining the demand assumption for the Alaskan Golfpurchase decision. Note that both the Name: and the Units: can be specifiedwith a cell reference.


FIGURE 3.4 Defining the demand assumption for the Alaskan Golf purchase decision.

will be entered for you automatically. You cannot define an assumption in a cellthat holds an Excel formula. If you try to do so, Crystal Ball will give you an errormessage.

Defining Profit as a Forecast Cell

A stochastic output cell in a Crystal Ball model is called a forecast cell. To definecell C11 as a forecast cell, follow the steps below:

1. Select cell C11, then click on Cell→Define Forecast.2. Enter the name and units (optional) in the text boxes (Figure 3.4).3. Click OK.

Cell C11 will turn blue to indicate that a forecast cell is defined there. Whenyou run the simulation, the forecast values will be calculated and collected for eachrun, then displayed graphically in a forecast window.

RUNNING CRYSTAL BALL

It is usually a good idea at this point to run just one iteration of the simulationto make sure that the calculations appear to be correct. To run one iteration, clickRun→Single Step, then make sure that the model correctly reflects the new valuesshowing in the assumption cells. This is good practice to help debug the logic of themodel.

When you are ready to run the model, click Run→Run, or just click theRun icon on the Crystal Ball toolbar. The forecast window should appear auto-matically. When the simulation stops you can use it to analyze the outputas shown in Chapter 2. Your forecast chart should resemble that shown inFigure 3.5.


FIGURE 3.5 Profit forecast for the Alaskan Golf purchase decision. Note that the probability ofnegative profit is less than 5 percent.

SOURCES OF ERROR

Monte Carlo simulation modeling is similar to statistical sampling in many respects,but differs in the type of up-front creativity required by the analyst. To compare andcontrast statistical sampling and simulation modeling, Figure 3.6 shows a stylizedview of steps taken in conducting a statistical study and a simulation study.

In a statistical study, the analyst defines a population to study, specifies numericalmeasures to be obtained, and develops a sampling frame—a list of all potentialpopulation elements that could potentially be included in a sample. The statisticalanalyst then randomly selects a sample of elements from the sampling frame andcalculates sample statistics such as the mean and standard deviation. These samplestatistics are then used to make inferences about the population using proceduressuch as hypothesis tests or confidence intervals. Virtually every introductory statisticstextbook covers these inferential statistical procedures. Note that no matter whatthe stated population of interest, it is the sampling frame that actually defines thepopulation about which inferences can be made in a statistical study.

In a simulation study, the analyst builds a model to represent the businessproblem of interest, runs the simulation, analyzes the output, then makes adecision—which can sometimes be simply to improve the model and rerunningseveral times before making a decision about the business problem. Crystal Ball isdesigned to run the simulations and provide graphical and numerical summaries


Statistical Study

Simulation Study

DefinePopulation

BuildModel

RunSimulation

AnalyzeOutput

MakeDecisions

SelectSample

CalculateStatistics

MakeInference

FIGURE 3.6 Stylized depictions of a statistical study and asimulation study.

of the output (forecasts) automatically and flawlessly. However, no matter whatthe stated business problem, it is the model that defines the problem about whichdecisions can be made in a statistical study.

Statistical analysts are concerned with both sampling and nonsampling error.Sampling error stems from the fact that sample statistics will differ from populationparameters simply by chance. Well-established theory tells us all we need to knowto control sampling error in a well-executed statistical study. Nonsampling erroris more insidious, coming about from faulty methods of data collection, incorrectsampling methods, errors in recording data, or erroneous entry of the data into thecomputer. It is difficult or perhaps even impossible to find and eliminate all sourcesof nonsampling error, although many professional statistical consulting firms havedevised methods to eliminate most sources of nonsampling error based on longexperience.

Crystal Ball is well debugged and tested, so there is no worry about the accuracyof its algorithms. By far, the biggest concern in a simulation study is model error.Model error is present because any model is an abstraction that cannot take intoaccount every detail and nuance of the real problem. By definition, a model containsonly the stochastic relationships that are most germane to the problem. All modelsare wrong to the extent that they omit minor details. Good models are useful ifthey include enough of the major details to help a decision maker reach a properlyinformed decision.

It is easy to create a bad model. Model risk is the risk of using the wrong model,implementing the correct model incorrectly, failing to assess the stochastic inputsadequately, or omitting important stochastic inputs that should be included. Goodmodel building is a craft that requires some experience to develop. Fortunately, thiscraft can be learned by studying the models in this book and those available fromother sources, such as the Web site www.crystalball.com.

Simulation error is caused by the fact that simulation is a sampling experiment.This is the same as sampling error in a statistical study. Simulation error should notbe ignored, but is usually a lesser problem than model error because it is relativelyinexpensive to reduce simulation error by simply increasing the number of trials.


It can also be measured and controlled through the variance reduction techniquesdescribed in Appendix C.

CONTROLLING MODEL ERROR

Good models have three basic requirements:

1. Verification. Ensuring that the values and formulas are entered correctly inExcel.

2. Validation. Assuring that your model faithfully mimics the actual situation orsystem.

3. Credibility. Acceptance of your verified and validated model for use by deci-sion makers.

Verification should be done at every step along the way. The Excel auditing toolscan be very useful for this purpose, and you should strive to make your models asreadable as possible so that you and others can understand the logic of the modelwhen you come back to it after setting it aside for some time.

Validation is done using a verified model. A validated model will give outputsthat seem to be reasonable to a subject matter expert. Proper validation of a modelis part of the craft of modeling that takes some experience to learn.

A credible model is obtained when a verified and validated model is acceptedfor use by the decision makers. Credibility is earned over time, and it is usually bestto keep decision makers informed about your progress in building the model, andto solicit their input at several points during the model building process. This willhelp get ‘‘buy-in’’ from them when it comes to put the model to work.

This chapter describes the basic steps involved in building models. As you gothrough the models in the rest of the book, you will begin to get a feel for how bestto go about creating models for your own purposes. The next chapter takes a morein-depth view of selecting and specifying Crystal Ball assumptions.

CHAPTER 4Selecting Crystal Ball Assumptions

T his chapter reviews basic concepts of probability and statistics using graphicsfrom Crystal Ball’s distribution gallery, a portion of which is shown in Figure 4.1.

If you have not had a class in basic probability and statistics at some point in yourlife or you need a refresher on these topics, consult a business statistics textbooksuch as Mann (2007). This chapter is intended to show the basics of how to specifyprobability distributions to be used as stochastic assumptions with Crystal Ball.

Version 7.2 of Crystal Ball has 20 distributions from which to choose whendefining assumptions. To see them, click the All button at the upper left of thedistribution gallery. Six basic distributions are described here along with the binomialdistribution.

CRYSTAL BALL’S BASIC DISTRIBUTIONS

Yes-No

Probabilists named the Bernoulli distribution in honor of the mathematician whoshowed analytically around 1700 the truth of the intuitive notion that when a faircoin is tossed repeatedly, it will come up heads about 50 percent of the time. Itis perhaps the simplest of all probability distributions. The random variable Y hasthe Bernoulli distribution if it can take only one of two possible values, y = 0 ory = 1. The value y = 1 is called a ‘‘success,’’ and y = 0 is called a ‘‘failure’’ inprobability parlance. In Crystal Ball, the Bernoulli distribution is known as theyes-no distribution.

Crystal Ball calls y = 1 ‘‘yes’’ and y = 0 ‘‘no’’ because these terms often makesense in a modeling context. For example, Figure 4.2 shows Crystal Ball’s yes-nodistribution for Pr(yes) = 0.5, where y represents the number of heads obtained inone toss of a fair coin. ‘‘Yes’’ means a head was tossed so y = 1, while ‘‘no’’ meansa tail was tossed so y = 0.

Now consider the type of situation that drew Bernoulli’s interest. The spreadsheetsegment in Figure 4.3 shows a simple model to be used for finding the number ofheads observed when tossing a fair coin five times. Each of the assumptions incells B3:B7 are yes-no distributions with Pr(yes) = 0.5, so each assumption cell will

36

Selecting Crystal Ball Assumptions 37

FIGURE 4.1 The basic distributions listed in Crystal Ball’sdistribution gallery.

contain 1 on approximately 50 percent of the trials and 0 on the remaining trials.Each assumption cell’s value is generated independently of the other cells’ values.The forecast in cell B8 has the formula =SUM(B3:B7).

Of course, we need not use simulation to model this situation because it is easyto determine the forecast distribution analytically. However, simulating a situationfor which we know the analytical solution can be comforting. If we get resultswith simulation that are in accord with the analytical results, then we have someassurance that simulation will provide good approximate answers to questionsregarding situations where analytical results are difficult or impossible to attain.

For a simple example of finding an analytical result, consider the spreadsheetmodel FiveTosses.xls shown in Figure 4.4, which shows each of the 25 = 32combinations of 0s and 1s that can occur on five tosses of a fair coin. Eachcombination is equally likely to occur. The number of heads in each combination is


FIGURE 4.2 Yes-no distribution to represent getting a head (y = 1) on one toss ofa fair coin.

FIGURE 4.3 Spreadsheet segment showing model fordetermining the distribution of five flips of a fair coin.Cells B3:B7 are yes-no(0.5) assumptions, and their sum incell B8 is a Crystal Ball forecast.


FIGURE 4.4 Spreadsheet segment showing model fordetermining the distribution of five flips of a fair coin.Cells B3:B7 are yes-no(0.5) assumptions, and their sum incell B8 is a Crystal Ball forecast.

found by summing across the row for each combination. So to find the probability ofeach of the possible numbers of heads, we simply divide the frequency of occurrenceof {0, 1, 2, 3, 4, 5} by 32, the total number of combinations to get the probabilitieslisted in cells C11:C16 in Figure 4.4. These are the probabilities associated with thebinomial(0.5,5) distribution used below.

Binomial

While not included in Crystal Ball’s distribution gallery list of basic assumptions,the binomial distribution is so closely related to the yes-no distribution that it isincluded here and used later in the chapter. The binomial(p,n) is the distribution of


FIGURE 4.5 Binomial(0.5,5) distribution to represent the number of heads on five tosses of a fair coin.

the sum of a fixed number, n, of Bernoulli trials that all have the same probability ofsuccess, p. Thus, the problem of determining the distribution of the number of headsin five tosses of a fair coin can be solved by using one Crystal Ball assumption—thebinomial(0.5,5) assumption shown in Figure 4.5.

Figure 4.6 depicts a model that gives the same results as that in Figure 4.3 byusing Crystal Ball to simply generate the number of heads in five tosses from thedistribution in Figure 4.5, and displaying the results in the forecast defined in cell B4with the Excel formula =B3. The forecast distribution in Figure 4.6 looks almostidentical to the forecast distribution in Figure 4.3, because the differences are dueonly to sampling error.

Discrete Uniform

The discrete uniform(L,H) distribution assigns equal probability to the set of integersbetween L and H, inclusive. For L = 1 and H = 6, it is the probability distributionrepresenting the number of spots showing on the top face of a fair die rolledrandomly. To illustrate the use of the discrete uniform, consider a problem withwhich Sir Isaac Newton dealt in the seventeenth century (Andel 2001).

The problem can be stated as follows:

■ Player A has 6 fair dice and wins if he rolls at least one ace (one spot showingon the top face of a die).

■ Player B has 12 fair dice and wins if he rolls at least two aces.■ Player C has 18 fair dice and wins if he rolls at least three aces.

Which player has the greatest chance of winning?


FIGURE 4.6 Simple model to represent the number of headsobserved on five tosses of a fair coin. Cell B3 is a binomial(0.5,5)assumption. Cell B4 is a forecast cell with the formula =B3.

Most seventeenth-century gamblers felt that because the ratio of rolls to aces(6:1) is the same for each player, the probability of winning should also be the samefor each player. Newton’s analytical solution to this problem uses the Binomialdistribution and is now considered trivial by probabilists. However, we will usesimulation to find the approximate values of each player winning and compare theresults to Newton’s analytical solution. Figure 4.7 shows a spreadsheet model of thesituation.

The Newton.xls model uses 6 discrete uniform assumptions in cells B5:B10,12 discrete uniform assumptions in cells E5:E16, and 18 discrete uniform assump-tions in cells H5:H22 to simulate the result of rolling each die. All 36 of theseassumptions resemble the distribution shown in Figure 4.8 for cell B5, which isdiscrete uniform on the integers {1, 2, 3, 4, 5, 6}. Each of the 36 discrete uniform dis-tributions generates observations independently of the others during the simulationruns.

In the cell to the immediate right of each die’s result in Newton.xls is an ExcelIF function that checks to see whether the result is a ace or not. For example, cell C5contains the command =IF(B5=1,1,0), which puts a 1 in cell C5 if the assumption inB5 delivers a 1 during any iteration, and a 0 otherwise. This is an example of anindicator variable, which is a useful modeling concept that we use often throughoutthis book. Cells C11, F17, and I23 find the sums of the indicator variables in thecells directly above them, C5:C10, F5:F16, and I5:I22, respectively. The cells


FIGURE 4.7 Spreadsheet segment showing model forNewton’s dice problem.

FIGURE 4.8 Discrete uniform distribution used for modeling the roll of one die forNewton’s dice problem.


TABLE 4.1 Table of values for Index in the Crystal Ballfunction =CB.GetForeStatFN(Range,Index) and thecorresponding forecast statistic. See Chapter 2 for adefinition of each statistic.

Index Statistic

1 Trials2 Mean3 Median4 Mode5 Standard deviation6 Variance7 Skewness8 Kurtosis9 Coefficient of variability10 Minimum11 Maximum12 Range13 Mean standard error

labeled A Wins? B Wins? and C Wins? are indicator variables to detect when thenumber of aces for A is greater than or equal to one, the number of aces for B isgreater than or equal to two, and the number of aces for C is greater than or equalto three, respectively. These cells, C12, F18, and I24 are defined as Crystal Ballforecast cells.

Finally, cells C13, F19, and I25 use the =CB.GetForeStatFN(Range,Index)command to find the mean of each forecast. The arguments for this command areRange, which is simply a reference to a Crystal Ball forecast cell, and Index, whichis an integer between 1 and 13. Specify the integer for Index that corresponds tothe desired forecast statistic listed in Table 4.1. For example, we use Index = 2 inNewton.xls because we want the means of the indicator variables in cells C13, F19,and I25.

The resulting means after 10,000 runs are shown in cells C13, F19, and I25to be 0.6628, 0.6205, and 0.5933. These values can be compared to the knownprobabilities obtained with the binomial distribution: 0.6651, 0.6187, and 0.5973,respectively. Thus, the solution to the problem is that Player A has the greatestchance of winning, followed by Player B, then by Player C. In a later chapter, wewill see how to determine the precision of the estimates from simulation models.

Uniform

The uniform distribution is the simplest of all continuous probability distributions.It has only two parameters, the minimum and maximum values. It produces anycontinuous value between the minimum and maximum with equal likelihood. The


FIGURE 4.9 Continuous uniform distribution used for modeling a firm’s revenue, where the minimumpossible value is $90, the maximum is $110, and all values in between are equally likely to occur.

uniform distribution depicted in the dialog window shown in Figure 4.9 models asituation where a firm’s revenues range from $90 to $110, and all values in betweenare equally likely to occur.

In dialog windows for its discrete distributions, Crystal Ball displays the possiblevalues of the random variable on the horizontal axis and the associated probabilitieson the vertical axis, as in Figure 4.8. For continuous distributions such as theuniform, Crystal Ball does not display values on the vertical axis because probabilityfor continuous random variables is associated with intervals on the horizontal axisand not with single values. Because they represent probabilities for intervals ratherthan single numbers, the plots for continuous distributions are graphs of probabilitydensity functions, or simply PDFs.

Use the uniform distribution when you know the minimum and the maximumvalues, but not a most likely value. The uniform distribution is completely spec-ified by its two parameters, Minimum and Maximum. Because all values betweenMinimum and Maximum are equally likely to occur, its PDF has a uniform heightover that range.

The uniform is sometimes called the ‘‘distribution of maximum ignorance,’’ andshould be replaced with a better estimate if one becomes available in later stages ofthe modeling process. However, there are some situations where the uniform maybe the best distribution; for example, to model (1) where a leak might occur on apipeline, or (2) time to failure of a component after a ‘‘burn-in’’ period, but beforethe required time to replace it.

The spreadsheet segment in Uniform.xls shown in Figure 4.10 models thesituation where a firm’s revenues follow the uniform(90,110) distribution and


FIGURE 4.10 Spreadsheet model for situation where a firm’srevenues are modeled as uniform(90,110), and where expensesare modeled as uniform(40,60). The resulting distribution ofprofit is triangular(30,50,70).

expenses follow the uniform(40,60) distribution. The difference, profit, is defined asa forecast in cell B5, and a forecast chart for it has been copied and pasted ontothe spreadsheet. It can be shown with the mathematical method of convolution(e.g., see Vose 2000) that the theoretical distribution of profit in this example istriangular(30,50,70), which is verified by the forecast chart in Figure 4.10.

Triangular

The triangular distribution is appropriate for use when you have little or no dataavailable, but you know the minimum, maximum, and most likely values of arandom variable. The triangular distribution is completely specified by its threeparameters, Minimum, Likeliest, and Maximum. These three values are sufficient todetermine the triangular shape shown in the icon. Of course, Minimum must be lessthan Maximum, and Likeliest must be in between (or equal to one of) these values.Figure 4.11 depicts a triangular(90,100,110) distribution.

The spreadsheet segment in Triangular.xls shown in Figure 4.12 models thesituation where a firm’s revenues follow the triangular(90,100,110) distributionand expenses follow the triangular(45,50,55) distribution. The difference, profit, is


FIGURE 4.11 Triangular(90,100,110) assumption used for modeling revenue inFigure 4.12.

FIGURE 4.12 Spreadsheet model for situation where a firm’srevenues are modeled as triangular(90,100,110), and whereexpenses are modeled as triangular(45,50,55).


defined as a forecast in cell B5, and a forecast chart for it has been copied and pastedonto the spreadsheet. Note how the distribution for profit has a bell shape similarto the normal distribution to be discussed next.

Because it is often used as a first estimate of the distribution, the triangulardistribution is applicable to many situations. When using it, you may wish to consultwith a subject matter expert (e.g., an engineer, cost analyst, or project manager) todetermine which values to use for the parameters.

Compared to the normal distribution, the triangular distribution over-emphasizes the tails and underemphasizes the middle values. Always try to replaceTriangular distributions with better estimates if they become available in later stagesof the modeling process.

Normal

The normal distribution is perhaps the most widely known continuous probabilitydistribution because it describes many natural phenomena. It has the familiar bellshape that Crystal Ball uses for its Define Assumption icon.

The normal distribution is specified by its two parameters, the Mean andStd Dev (standard deviation). Because it is symmetrical, the mean is equal to themedian (50th percentile). The mode (point on the horizontal axis at which the PDFis highest) is also equal to the mean and median. Values simulated from the normaldistribution are more likely to be close to the mean than far away. Figure 4.13 showsa normal distribution with mean 10 percent and standard deviation 5 percent.

Some examples of what might be modeled by a normal distribution include (1)the rate of return on stocks, (2) the rate of inflation, (3) sales revenue, (4) heights of

FIGURE 4.13 Normal(10%, 5%) distribution.


FIGURE 4.14 File CLT.xls built to demonstrate the effects of the Central Limit Theorem. Each day’ssales is generated independently from one of the binomial(0.75,3) assumptions in Cells B4:B33. Thisassumption appears at the bottom of the spreadsheet. Note that rows 12 through 31 are hidden.

people, or (5) time to complete work composed of many individual tasks, to namea few possible applications. A well-known mathematical result—the Central LimitTheorem (CLT)—explains why the normal distribution does an adequate job ofdescribing many natural phenomena. Not every random variable encountered whenbuilding Crystal Ball models is normally distributed, but the Normal often workswell as a first model for many stochastic assumptions.

Central Limit Effect The central limit effect is what causes the normal distributionto be a suitable choice for modeling many natural phenomena. The model inFigure 4.14 and the forecast windows in Figures 4.15, 4.16, and 4.17 illustrate thiseffect.

The file CLT.xls generates daily sales with a binomial(0.75,3) distribution,a skewed distribution that is far from normally distributed, as can be seen in


FIGURE 4.15 Distribution of sales for one day. Each day’s sales isgenerated from the binomial(0.75,3) distribution, and this forecast chartdepicts that distribution.

FIGURE 4.16 Distribution of sum of seven days’ sales. Each day’s sales isgenerated from the binomial(0.75,3) distribution. This forecast chartshows how the distribution of weekly sales is bell-shaped, but not quite anormal distribution.


FIGURE 4.17 Distribution of sum of 30 days’ sales. Each day’s salesis generated from the binomial(0.75,3) distribution. This forecastchart shows how the distribution of monthly sales is close to normalbecause of the central limit effect. The curve superimposed on thehistogram is the PDF for the normal distribution with mean andstandard deviation parameters that are equal to the sample mean andstandard deviation statistics calculated from the 10,000 simulatedvalues.

Figure 4.15. However, when we look at the distribution of weekly sales, which isthe sum of seven days’ sales, we see the mound-shaped distribution that is shown inFigure 4.16. The distribution of monthly sales, the sum of 30 days’ sales depicted inFigure 4.17, very much resembles the bell-shaped probability density function (PDF)of the normal distribution, which is superimposed on the frequency chart of theforecast values for comparison.

In financial modeling, the random variables of interest often are measuresproduced as sums of other random variables. For example, the normal distributionis often used to model rates of return. This makes it easy to analyze risks associatedwith a single period without using Crystal Ball. For example, if we know that therate of return on Stock A is normally distributed with mean 10 percent and standarddeviation 15 percent, then the probability of a negative return is 0.2525. This canbe seen in Figure 4.18, where the probability of a negative return is found in cell B5with the command =NORMDIST(0,B3,B4,TRUE). The file also uses Crystal Ball with10,000 iterations find an approximate value of 25.23 percent for this probability.

Mixture of Normals Sometimes you may be interested in modeling a situation wherea stochastic input is a mixture of two distributions. For example, suppose you


FIGURE 4.18 This file uses Excel’s NORMDIST distributionfunction and Crystal Ball to find the probability that the rateof return is negative for a stock with mean return of 10percent and standard deviation of 15 percent.

are interested in simulating stock rates of return for a market in which one oftwo regimes will prevail: Regime 1, where monthly rates of returns are normallydistributed with mean µ = 1 percent and standard deviation σ = 1 percent, andRegime 2, where monthly rates of returns are normally distributed with mean µ = 1percent and standard deviation σ = 3 percent. The market is in Regime 1 on 80percent of the months and in Regime 2 on 20 percent of the months. See Figure 4.19and the file Mixture Model.xls.

In the mixture model simulation depicted in Figure 4.19, a yes-no assumptionis used in cell A9 to generate values of either 0 or 1. The prevailing regime isdetermined from the yes-no assumption with the formula =2-A9. Thus, when theyes-no assumption cell is 1, cell B9 will indicate Regime 1, and when the yes-noassumption is 0, cell B9 will indicate Regime 2. Because a value of 1 is generated onabout 80 percent of the trials as shown in cell B4, Regime 1 prevails about 80 percentof the time and Regime 2 prevails on the rest of the trials. The normally distributedrate of return is generated in cell E9 from an assumption whose parameters dependon the prevailing regime. The total return in cell A11 is mound-shaped but hasheavier tails than a normal distribution.


FIGURE 4.19 Mixture model for generating rates of returnon a stock under two different regimes. The forecast windowpasted at the bottom of the spreadsheet has the PDF of anormal distribution superimposed on it to show how themixture of two normal distributions is mound-shaped but hasa higher peak and heavier tails than a single normaldistribution with mean and standard deviation parametersthat are equal to the sample mean and standard deviationstatistics calculated from the 10,000 simulated values.

Lognormal

Unlike the normal distribution, the lognormal distribution is bounded on the leftby zero; however, it is unbounded on the right just as the normal distribution. Thismakes it useful for situations where values are positively skewed and cannot benegative, such as the total return on stock when the stockholder’s potential loss islimited to the amount he or she has invested, or for sales of a product, which cannotbe negative.

The lognormal distribution takes its name from the fact that it represents arandom variable whose natural logarithm follows the normal distribution. Like the


FIGURE 4.20 Model of cumulative effect of growth of $100 over a20-year period where each year’s return is a lognormally distributedrandom variable with mean 1.1734 and standard deviation 0.3607 incells E1 and E2, respectively. Note that rows 6 through 22 are hidden.

normal distribution, it has two parameters, Mean and Std. Dev. File Lognormal.xlsin Figure 4.20 has a model where each year’s return is a lognormally distributedrandom variable with mean 1.1734 and standard deviation 0.3607 in cells F1 andF2, respectively. (Notice that if you click in the Mean field you will see that themean is defined as an absolute reference $F$1. This facilitates copying and pastingCrystal Ball data from one assumption cell to another. The Std. Dev. is defined inthe same way, which you can verify by clicking in that field.) The model generatesannual total returns independently each year from the same lognormal distribution,and the forecast in cell C24 shows the potential distribution of wealth at the end ofYear 2026. This distribution appears in Figure 4.21.

Many variables in financial modeling are suitable for use of the lognormaldistribution; for example, stock or real estate prices, critical pharmaceutical doses,salaries in a company, amount of oil in a reservoir, or incubation time for aninfectious disease. This comes about from the central limit effect where randomvariables arise as products of other variables. Products can be found as sums oflogarithms, so the CLT implies that the sum of the logarithms will be normal. Ifthe logarithm of a random variable is normally distributed, then the variable islognormally distributed.


FIGURE 4.21 Forecast window from model shown in Figure 4.20 of cumulative effect of growth of$100 over a 20-year period where each year’s return is a lognormally distributed random variable withmean 1.1734 and standard deviation 0.3607 in cells F1 and F2, respectively.

USING HISTORICAL DATA TO CHOOSE DISTRIBUTIONS

In the Lognormal.xls model, we used historical data to estimate the mean andstandard deviation for future returns. These data are located on the Data worksheetof the Lognormal.xls file. If you have historical data on an input variable, there areat least two different methods for using them in a Crystal Ball model: (1) directsampling, and (2) sampling from a fitted distribution.

Direct Sampling

This method uses the data values directly in the simulation. For example, we cancalculate the historical annual returns on a stock and use them for projecting futurereturns. This is illustrated in the file DirectSampling.xls shown in Figure 4.22, whichhas historical total returns for small cap stocks for each year between 1926 and2002 inclusive.

The assumptions in cells B5:B23 are identical, but generate independent obser-vations from a Discrete Uniform distribution on the integers {1926, 1927, . . . ,2002}. These integers correspond to the rows of the array in cells A2:B80 on theData worksheet, which are used with Excel lookup commands in Cells C5:C23 on


FIGURE 4.22 Crystal Ball model todemonstrate how to use directsampling of historical data forpredicting future returns. Each ofcells B5:B23 has a discreteuniform(1926,2002) distributionrepresenting the years of the returndata in a separate worksheet. Eachtime a year is selected randomly incolumn B, the corresponding return isplaced in the same row of column C.

the Model worksheet. For instance, cell C5 has the command

=VLOOKUP(B5,Data!$A$2:$B$78,2,FALSE)

which takes the randomly generated year from cell B5 and finds the correspondingreturn to put in cell C5. This method is equivalent to writing each return on a slipof paper and placing in a bowl, then sampling with replacement from the bowl todetermine the return for each year.


The downsides to the direct sampling approach are that

■ The simulation can only reproduce what has already happened, and■ The number of trials usually exceeds the number of data values available, so

that you will be using the same values many times over.

Thus, using direct sampling can lead to a false sense of precision and is not generallyrecommended.

Sampling from a Fitted DistributionIn this method, Crystal Ball uses standard techniques of statistical inference tofit a theoretical distribution to your data using one of the distribution gallery’scontinuous distributions. The fitting and selection is nearly automatic, although itdoes require some judgment and subject matter knowledge to use most effectively.

If a suitable theoretical distribution can be found, sampling from a fitted distri-bution is preferred over direct sampling or sampling from an empirical distributionbecause:

■ Historical datasets typically contain relatively few observations. Fewer than 100is not uncommon in some finance applications (as in the example we use here),and so are ‘‘rough.’’ A fitted distribution will typically be ‘‘smooth’’ and mightwell better represent the underlying stochastic process generating the data thandoes the direct sample from a limited number of past observations.

■ Unless extra tail information is appended to the historical data, it cannot generatevalues less than the minimum nor greater than the maximum. In many models,the tails of the distribution produce values that lead to some of the model’s mostinteresting results and thus provide useful information. For example, in ‘‘stress-testing’’ a portfolio, analysts evaluate the impact of potential future occurrencesof events that could cause problems in a portfolio. Such stress scenarios oftencome from the tails of the input distributions affecting the portfolio.

■ There is often good reason to expect that a theoretical distribution is applicablein many financial applications. For example, many researchers have found thatannual stock returns often appear to be normally distributed, and stock pricesgenerally follow lognormal distributions.

■ Fitted distributions are efficient. Direct sampling requires storing all n obser-vations for reuse. Sampling from a fitted distribution is accomplished for allof Crystal Ball’s continuous distributions with algorithms that its authors haveembedded in the program code.

■ Fitted distributions are easier to change. For example, if the historical volatilityof a stock is 20 percent, but you think it is likely to double, all you have to dois change the standard deviations of the assumptions you are using to generatereturns.

In situations for which we know that there is a limit on how large or small agenerated value can be, we can set a limit on our generated values for any distributionwith a truncation limit in Crystal Ball.


FIGURE 4.23 Dialog for step 1 of fitting a distribution to data.

Fitting Distributions to Data

Crystal Ball provides the button Fit. . . in the distribution gallery window to fit asingle distribution to a source of data, and Run→Tools→Batch Fit. . . to fit morethan one distribution to multiple sources of data.

Follow these steps for an example of how to fit a distribution to empirical datawith Fit. . .

1. Open the file Lognormal.xls. Select the Data worksheet, then click on cell C2.Choose Define→Define Assumption. . . from the top menu or click on the defineassumption icon on the Crystal Ball toolbar. You should see a dialog box likethat shown in Figure 4.23.

2. Click on the Fit. . . button at the bottom right center of the dialog box inFigure 4.23. Enter the range B2:B78 into the Range: field of the dialogbox as shown in Figure 4.24. For this example, choose the radio buttonsto select All continuous distributions and the Anderson-Darling ranking methodas shown in the Fit Distribution dialog in Figure 4.24. Then click the OK


FIGURE 4.24 Dialog for step 2 of fitting a distribution to data.

button on the Fit Distribution dialog. You will get a comparison chart simi-lar to that shown in Figure 4.25. By clicking on Next >> and << Previous,you can see plots of the histogram of the data and the theoretical distribu-tion functions for the various fitted distributions. The Cumulative Frequencyor Reverse Cumulative Frequency views provide better comparisons of thehistogram to the theoretical distribution than does the Frequency view.

3. Click << Previous or Next >> until you see Fit #7: Lognormal. Then clickAccept and you will see the dialog shown in Figure 4.26. This assumptioncan be copied and pasted to other cells in the model such as Cells B5:B24 onthe Model worksheet in Lognormal.xls using the Crystal Ball Copy Data andPaste Data commands.

Goodness-of-Fit Testing Crystal Ball has built into its distribution-fitting procedurethe algorithms to estimate parameters and assess the goodness of fit between theempirical distribution function (EDF) of your dataset and the cumulative distributionfunction (CDF) of each applicable continuous distribution in its distribution gallery.Not all of the distributions in the gallery are applicable to all datasets. For example,a lognormal distribution will not be fit to a dataset containing negative valuesbecause the support of the lognormal distribution is bounded from below by zero.Depending on your data, you may see a message from Crystal Ball warning you ofthis fact.


FIGURE 4.25 Comparison chart for fitting a distribution to data.

FIGURE 4.26 Lognormal distribution fit to data in file Lognormal.xls.


This section gives a brief description of the procedures used by Crystal Ball forfitting distributions. Suppose a random sample of size n is generated by a stochasticprocess governed by a cumulative distribution function F(x) (the CDF). Intuitively,the goodness of fit can be tested by measuring the ‘‘closeness’’ of the EDF, Fn(x), tothe theoretical CDF, F(x).

Eyeball Test One of the best ways to assess goodness of fit is simply to use the‘‘eyeball test,’’ by comparing the plots of the EDF and each candidate CDF. CrystalBall helps you do this with frequency charts and cumulative frequency charts as partof its fitting procedure. By specifying the Cumulative Frequency view and clickingon Next >> and << Previous as described above for the data in Lognormal.xls,you can use the eyeball test to assess the fit between the EDF of your data and theCDF of each of Crystal Ball’s applicable distributions.

For those who prefer a quantitative measure of goodness of fit, Crystal Ballalso calculates the values of three test statistics under the hypothesis that the datawere generated from its applicable theoretical, continuous distributions. Figure 4.27shows the goodness-of-fit statistics calculated for the data in Lognormal.xls. InCrystal Ball Version 7, the p-values for these test statistics are not provided, butthey may be found or estimated using a reference book such as D’Agostino andStephens (1986) or an advanced textbook on statistical analysis. These statistics areused for ordering the distributions according to the ranking method you specify inthe dialog window shown in Figure 4.24. For example, we chose to rank with the

FIGURE 4.27 Goodness-of-fit statistics for data in file Lognormal.xls.


Anderson-Darling statistic and the lognormal distribution appears as the seventhentry in the table shown in Figure 4.27.

Do not assume that the highest ranking distributions in the table necessarilycorrespond to the best distributions for use in your model. The choice of the bestdistribution depends on its use in your model and the judgement of a subject matterexpert (SME). For example, while the logistic distribution is ranked highest inFigure 4.27, the lognormal distribution is a better choice for total return in this casebecause of the product version of the central limit effect described above.

Caveats Things to keep in mind when using historical data include:

■ Empirical data are usually not available, at least not immediately. For newofferings of products or services, there may well be no historical data at all, andfor improved offerings of products or services, the historical data may not beindicative of what is to come in the future. In situations like these, you will needto specify assumptions based on subject matter expert input. Of course, oncethe new offering is in place, you should have processes in place to record dataon the stochastic variables of interest for reparameterizing your model when itis updated.

■ Some information may be biased or otherwise inappropriate, simply because thedata collector did not anticipate that the data would be used for risk analysisor financial modeling. Thus, sometimes the data may be a mix of observationsfrom two or more stochastic variables, or be collected over a period of timewhere the parameters (for example, the mean or standard deviation) for a singlevariable may have changed over time. This may make the historical data invalidfor use in fitting distributions.

■ As mentioned above, fitting a distribution to data requires judgment to determinewhether the ‘‘best-fit’’ distribution is appropriate. The ability to make thisjudgement is developed with experience. Novices may have to rely on SMEs fortheir judgement.

The good news is that the results of interest in most models often depend on themean and the variance of inputs more than on the specific probability distributionsused. If you find yourself in a situation where potential users of your model arequestioning the appropriateness of the input distributions, you may find it helpfulto try different distributions. As long as the different distributions have the samemean and variance, the central limit effect will apply to most realistic models and theforecast distributions will be relatively insensitive to the choices of the distributionfamilies of the assumptions.

Paradoxically, while good empirical data are the best source for helping todetermine which assumptions to choose from the distribution gallery, you shouldnot rely on them too much. Subject matter knowledge and good judgment are alsonecessary ingredients for constructing good models.


What If No Historical Data Are Available?

Financial models are often built to analyze situations that do not yet exist; forexample, new products or projects in which the company has little or no historicaldata to help choose assumptions. In this case, you will have to make your ownsubjective estimates of which assumptions to use in the model, or solicit thehelp of a SME. The uniform and triangular distributions are often good firstchoices.

The uniform distribution is sometimes called the ‘‘distribution of maximumignorance’’ because it requires only two parameters to specify, the Minimum andthe Maximum. By considering every value between the minimum and maximum tobe equally likely to occur, we are being very conservative in our approach. Quiteoften SMEs who know nothing about probability distributions will be able to helpyou choose the parameters of the uniform distribution simply by your asking for thehighest and lowest possible values that they think may occur.

Depending on the distribution, you may define the distribution using differentcombinations of the mean, standard deviation, minimum, maximum, or selectedpercentiles. For example, Figure 4.28 shows the alternative ways for defining alognormal distribution. The possible parameter sets that can be used depend on thedistribution to be defined. It is sometimes easier to elicit estimates of percentiles

FIGURE 4.28 Parameters drop-down menu for the lognormal distribution.


or other parameters for assumptions from SMEs, and the Parameters menu wasdeveloped to make easier the task of defining distributions. Expert opinion is usuallymore readily available than empirical data, but you should be prepared to justifyand/or defend your choices no matter how you selected your assumptions.

If you have a better feel for the situation than maximally ignorant, you maywant to consider using the triangular distribution. Like the uniform, it requires aminimum and maximum values, but it also requires a Likeliest value. This is alsoknown as the mode of the distribution and specifies the point on the horizontal axisabove which the density is the highest.

Building a model without historical data can provide many valuable insights.However, as soon as possible, you should collect data on the stochastic variablesdriving your forecasts and parameterize the assumptions. Among other reasons, thiswill also allow you to estimate any correlations between the assumptions, which canmake a huge difference in the forecast results.

SPECIFYING CORRELATIONS

Crystal Ball gives you the ability to specify correlations between assumptions in yourmodels. This is important because quite often several assumptions will be affected bythe same factor. For example, returns on stocks all tend to be affected by the samemarket forces, although to varying degrees. The correlation coefficient measures thedegree to which one assumption moves with another.

The type of correlation known to most people from their college statisticscourses is the Pearson product-moment correlation coefficient. This is the samecorrelation coefficient that Excel calculates with its =CORREL(array1,array2) com-mand. However, it turns out that the Pearson coefficient does not generalize easilyto all distributions, so Crystal Ball uses the Spearman rank correlation coefficientinstead of the Pearson. In this section, we elaborate on the distinction betweenPearson and Spearman correlation coefficients.

Pearson Correlation Statistic If we have n observations on a random variable Xdenoted as x1, x2, . . . , xn, and n observations on another random variable Y denotedas y1, y2, . . . , yn, we can calculate the degree of linear association between themwith the Pearson correlation coefficient, rxy. This coefficient ranges from −1 to+1. If X tends to be above (below) its mean value whenever Y is above (below)its mean value, then rxy will be close to +1. Conversely, if X tends to be below(above) its mean value whenever Y is above (below) its mean value, then rxy will beclose to −1. Simply put, the correlation coefficient is a measure of the tendency forone random variable’s observations to follow the movement of another variable’sobservations.

The Pearson correlation coefficient is calculated by first finding the mean of eachset of observations as

x =∑n

i=1 xi

nand y =

∑ni=1 yi

n.


FIGURE 4.29 Comparison of correlation coefficients fordata in PearsonSpearman.xls. Note that rows 6 through79 are hidden.

Then the Pearson product-moment correlation statistic is calculated as

rxy =∑n

i=1(xi − x)(yi − y)√∑n

i=1(xi − x)2∑n

i=1(yi − y)2.

The spreadsheet shown in Figure 4.29 illustrates the calculation of correlationsin Excel. If we denote as x1, x2, . . . , x79 the total returns on small cap stocksin cells B2:B80, and denote as y1, y2, . . . , y79 the total returns on large capstocks in cells C2:C80, then the Excel formula =AVERAGE(B2:B80) calculatesthe sample mean x, =AVERAGE(C2:C80) calculates the sample mean y, and=CORREL(B2:B80,C2:C80) calculates the Pearson correlation statistic, rxy.

Spearman (Rank) Correlation Statistic The Spearman, or rank, correlation statistic,rS

xy, is a nonparametric estimator of the correlation coefficient that is calculatedfrom the ranks of the observations. Crystal Ball uses Spearman correlations becausethe Pearson correlation does not work well in simulation algorithms for every oneof Crystal Ball’s distributions, but the Spearman correlation coefficient does. TheSpearman correlation coefficient is the Pearson correlation coefficient calculated forthe ranks of the observed values of X and Y.

The rank of an observation is simply its position number in an orderedarray when the data are sorted in ascending order. Note that by default, theExcel command =Rank(Number,Ref,Order) will calculate the rank of Number inthe (ordered or unordered) array Ref based on descending order if Order = 0or is blank. The discussion below is applicable whether the data are sorted inascending or descending order; however, for our purposes, it is easy to change toascending order by specifying Order = 1. Ascending order is used conventionallyby most statisticians in assigning ranks. For example, if the observed values ofX are

1, 7, 3, 1, 0, 1, 5, 2, 7, 3


when arrayed in ascending order they are

0, 1, 1, 1, 2, 3, 3, 5, 7, 7

With ties in ranks, assign the average rank to each tied value. Excel does not assignaverage ranks by default with its =RANK(Number,Ref,Order) function, but this canbe accomplished by using a correction given on an Excel help page:

=[COUNT(Ref) + 1 − RANK(Number,Ref,0)− RANK(Number,Ref,1)]/2.

This is illustrated in the table below:

i xi yi rank(xi) rank(yi)

1 1 7 3 9.52 7 3 9.5 3.53 3 6 6.5 7.54 1 7 3 9.55 0 6 1 7.56 1 5 3 67 5 4 8 58 2 3 5 3.59 7 2 9.5 1.510 3 2 6.5 1.5

Let X′ = rank(xi) and Y ′ = rank(yi). The Spearman rank correlation statistic isthe Pearson product moment correlation calculated from the ranks,

rSxy =

∑ni=1(X′

i − X′)(Y ′i − Y ′

i)√∑ni=1(X′

i − X′)2∑n

i=1(Y ′i − Y ′

i)2

The calculations are illustrated in the file PearsonSpearman.xls. If there are no tiesin the ranks, a shortcut method for calculating Spearman correlation is

rSxy = 1 − 6

∑ni=1 d2

i

n(n2 − 1),

where di = x′i − y′

i.

Using Crystal Ball to Calculate Correlations between TwoAssumptions

You can calculate Spearman correlations between two cell ranges using CrystalBall’s Correlate. . . button on the Define Assumption dialog window. Using the stepslisted below, this was done for the assumptions in cells H2 and I2 from the data in


cells B2:B78 and C2:C78, respectively, in the spreadsheet PearsonSpearman.xls.Cell H2 is lognormal(1.18,0.36), and cell I2 is lognormal(1.13,0.22).

The procedure for calculating the correlation between cells H2 and I2 follows.Once you go through this, you should be able to define correlation between any twopairs of assumptions for which you have data.

1. Select either cell H2 or cell I2, then click on Define→Define Assumption. . . Forthis example, I have selected cell H2, SC Stocks, as shown in Figure 4.30.

2. Click on the Correlate. . . button to get the Define Correlation dialog shownin Figure 4.31, then click on Choose. . . to get the Choose Assumptions dialogshown in Figure 4.32.

3. Click on the check box next to LC Stocks as shown in Figure 4.32, then clickOK. The Define Correlation dialog will appear.

4. Click on the Calc. . . button to get the Calculate Correlation Coefficient windowshown in Figure 4.33. Enter the cell ranges B2:B78 and C2:C78 as shown inFigure 4.33, then click OK. You will see the Spearman correlation coefficientvalue 0.81119, which is the same as the value shown in cell A81 in Figure 4.29.

5. Click OK to get back to the Define Assumption dialog, which now has(Correlated) added to the label in the dialog window title bar to indicatethat the assumption is correlated with at least one other assumption in thespreadsheet.

6. Click OK again to close the dialog.

FIGURE 4.30 Define Assumption dialog for cell H2 in file PearsonSpearman.xls. The Correlate. . .

button is second from right at the bottom of the dialog window.


FIGURE 4.31 Define Correlation dialog for cell H2 in filePearsonSpearman.xls.

FIGURE 4.32 Choose Assumptions dialog for cell H2 in filePearsonSpearman.xls.

Batch Fit

Crystal Ball’s Batch Fit feature enables you to fit distributions to and calculatecorrelations for several sets of data in one set of steps. In this section we will fitlognormal distributions and find correlations for monthly total returns observed


FIGURE 4.33 Calculate Correlation Coefficient dialog for cell H2 infile PearsonSpearman.xls.

FIGURE 4.34 First step in using Batch Fit with data in file BatchFit.xls.

during the period January 1998 through December 2004. These returns are in thefile BatchFit.xls.

Follow these steps to use Batch Fit:

1. Click on Run→Tools→Batch Fit. . . in the top menu. The Select distributions(step 1 of 3) dialog window will appear.


FIGURE 4.35 Second step in using Batch Fit with data in file BatchFit.xls.

FIGURE 4.36 Third step in using Batch Fit with data in file BatchFit.xls.


FIGURE 4.37 Output from using Batch Fit with data in file BatchFit.xls.

2. Move all distributions except lognormal from the Selected Distributions fieldto the Available Distributions field as shown in Figure 4.34. Move distributionsback and forth between these fields using the << and >> buttons found betweenthe two fields. With lognormal as the only selected distribution, click Next.

3. Specify the location of the data, punch the same radio buttons, and check thebox indicated in Figure 4.35, then click Next.

4. Specify the location of the output, punch the same radio buttons, and check theboxes indicated in Figure 4.36, then click Start.

Your spreadsheet should resemble Figure 4.37.Cells G5:I5 in Figure 4.37 are now Crystal Ball lognormal assumptions

having means, standard deviations, and correlations calculated from the data incells B4:D87. They can be copied to other cells in your spreadsheet using CrystalBall’s CopyData and PasteData commands.

CHAPTER 5Using Decision Variables

T he first four chapters covered the basics of specifying Crystal Ball assumptionsand analyzing Crystal Ball forecasts. This chapter covers the basics of defining

and using Crystal Ball decision variables and its decision support tools, DecisionTable and OptQuest.

DEFINING DECISION VARIABLES

Decision variables are spreadsheet cells in which the values are varied systemat-ically rather than sampled randomly, as are assumptions. They can be cells thathold values dictated by actual decisions or cells for which we just want to seethe effect of one or two variables on selected forecasts in a form of sensitivityanalysis.

As an example of the latter, consider the model depicted in Figure 5.1,TwoCorrelatedAssets.xls, where we have two correlated assets and we wish tovary the correlation coefficient to see the effect on the rate of return of a portfoliocomposed of the two assets. To keep things simple we will simulate a model forwhich we know the true answer so that we can compare the simulation results tothe truth.

Consider a portfolio composed of two assets, A and B. Asset A has a normallydistributed rate of return with mean, µA = 10 percent, and standard deviation,σA = 20 percent. Asset B has a normally distributed rate of return with mean,µA = 15 percent, and standard deviation, σB = 30 percent. We will invest half ofour available funds in Asset A and half in Asset B, and we are interested in seeinghow the distribution of the rate of return of our portfolio varies as a function ofthe correlation coefficient, ρ, between the rates of return on Assets A and B. InTwo Correlated Assets.xls, cells A9 and A10 are assumptions, and cell A13 is aforecast.

The assumptions in cells A9 and A10 of the file TwoCorrelatedAssets.xls aredefined as normal distributions as described in Chapter 4. However, to reflect thelimited liability of stock ownership, the rates of return on both assets are truncated

71


FIGURE 5.1 Spreadsheet model infile TwoCorrelatedAssets.xls, which models rateof return on portfolio as the correlationcoefficient, ρ, varies from −1.00 to +1.00 insteps of 0.25. Cell D5 is a decision variable,cells A9 and A10 are assumptions, and cell A13 isa forecast.

on the left at −100 percent. This is accomplished by entering this value in thefield directly above the Mean field as shown in Figure 5.2 for the rate of returnassumption for Asset B. With a mean of 15 percent and a standard deviation of30 percent, the normal distribution will rarely produce any values below −100%(which is more than 3.8 standard deviations below the mean), but truncating thedistribution ensures that on those rare occasions when a value less than −100%is generated from a nontruncated normal distribution for Asset B, Crystal Ball willdiscard it and generate another random value in its place. Asset A’s rate of return isalso bounded from below by −100%.

Note that the title bar of the dialog in Figure 5.2 for cell A10 indicates that it iscorrelated with another assumption, in this case the assumption in cell A9. Duringany simulation run, the correlation coefficient is the value in cell D5, which is whatwe will vary with the Decision Table tool.

To make Cell D5 into a decision variable, first click on it. Then selectDefine→Define Decision. . . from the top menu or click on the Define Decisionicon on the Crystal Ball toolbar. Fill in the fields as shown in Figure 5.3. Thistells Crystal Ball that you wish to vary the value in cell D5 from −1.00 to +1.00in discrete steps of 0.25. Click OK and cell D5’s background will turn yellow toindicate that a decision variable is defined in that cell. You are now ready to use theDecision Table tool or OptQuest with the model.

Using Decision Variables 73

FIGURE 5.2 Define assumption dialog for cell A10 infile TwoCorrelatedAssets.xls.

FIGURE 5.3 Define Decision Variable dialog window.

DECISION TABLE WITH ONE DECISION VARIABLE

Select Run→Tools→Decision Table. . . from the top menu. You will get a dialogwindow for step 1 like that shown in Figure 5.4. For models with more than oneforecast defined, you would first highlight the forecast for which you wish toanalyze the sensitivity to the decision variable. As there is only one forecast in thismodel, there is no choice to be made here, so click Next > to select the forecastPortfolio ROR.


FIGURE 5.4 Dialog window for step 1 of using Decision Table.

The dialog for step 2 is where you choose the decision variables to evaluate,as shown in Figure 5.5. Again, because there is only one decision variable defined,click >> to move the decision variable Correlation to the Chosen Decision Variablesfield, then click Next>.

Step 3 is where you specify the Decision Table options. Take the default optionsdepicted in Figure 5.6, then click Start. Crystal Ball will begin running nine sets of10,000 simulation trials, one set for each of the following values of the correlationcoefficient:

{−1.00, −0.75, −0.50, −0.25, 0.00, 0.25, 0.50, 0.75, 1.00}

After it has finished running 90,000 simulation trials, Crystal Ball will createa separate workbook like that in Figure 5.7 holding nine forecasts and threebuttons:Trend Chart, Overlay Chart, and Forecast Charts.

Trend Chart

To see the trend chart, select the forecasts in cells B2:J2, then click Trend Chartto get a graphical display similar that shown in Figure 5.8. A trend chart displaysthe certainty bands for several forecasts on one plot. The chart in Figure 5.8 clearly





FIGURE 5.7 Results of using Decision Table.

FIGURE 5.8 Trend chart displaying results of using Decision Table.

shows how the variability of the portfolio rate of return increases as the correlationcoefficient goes from −1.00 to +1.00. At the left end of the trend chart, the 90percent certainty band extends from 4.3 percent to 20.7 percent for the forecastPortfolio ROR (1), which is the forecast generated when the correlation coefficient is−1.00. At the right end, the 90 percent certainty band extends from −28.6 percentto 53.6 percent for the forecast Portfolio ROR (9), which is the forecast generatedwhen the correlation coefficient is +1.00. These correspond to the same certaintyintervals indicated in the forecast charts for Portfolio ROR (1) and Portfolio ROR (9)displayed in Figure 5.9.

Note how the certainty bands diverge from left to right, but remain centeredon 12.5 percent. They correspond very well to the theoretical bounds calculated inTwo Correlated Assets.xls, and displayed in Figure 5.10.


FIGURE 5.9 Frequency charts for Portfolio ROR (1) and (9) fromDecision Table results.

These charts conform well to theory, which holds that the rate of return onthe portfolio will be normally distributed with mean µP = wµA + (1 − w)σB, andstandard deviation

σP =√

w2σ 2A + (1 − w)2σ 2

B + 2w(1 − w)ρσAσB, (5.1)

where w is the weight (0 ≤ w ≤ 1) of Asset A in the portfolio. For this example,w = 0.5.


FIGURE 5.10 Theoretical 90 percent certaintybounds for trend chart displaying results of usingDecision Table. Bounds are calculated asµP + �−1(y)σP, where �−1(y) is the inverse CDF forthe standard normal distribution, y = 0.05 for thelower bound, and y = 0.95 for the upper bound.

It is worth noting that our model differed slightly in at least two ways fromthe theoretical model underlying Expression (5.1). First, Crystal Ball uses Spear-man correlation, ρS, for sampling instead of the Pearson correlation, ρ, used inExpression (5.1); however, for normally distributed rates of return this makes littledifference. Second, we truncated the rate of return assumptions on the left, whichhas the effect of changing the standard deviations slightly, but this also made littledifference.

Overlay Chart

The forecast charts in Figure 5.9 look similar, but the scales of their horizontal axesare much different. To make it easier to compare forecasts, use an overlay chart. Clickon cell B2, then hold down the Ctrl key and click on cell J2 to select the forecasts


FIGURE 5.11 Overlay chart displaying results of using DecisionTable.

for Portfolio ROR (1) and Portfolio ROR (9). Then click the Overlay Chart button toget a chart like that displayed in Figure 5.11, which clearly shows the difference indispersion between the forecasts for Portfolio ROR (1) and Portfolio ROR (9). Youcan create an overlay chart for more than two forecasts at a time, but do so withcare as they become hard to read when too many forecasts are included.

DECISION TABLE WITH TWO DECISION VARIABLES

You can also use the Decision Table tool with two decision variables. The output issimilar to the output with one decision variable, except that Crystal Ball will producean array containing a forecast for every possible combination of the values specifiedfor each decision. This section contains an example of using Decision Table with amodel built as an illustration of how to estimate a value for managerial flexibility.

ModelFor an example of a situation with two decision variables, suppose that a firm caninvest in a project having a three-year life and a terminal value that depends on thecash flow in the final quarter of the third year. Suppose further that there are onlytwo sources of uncertainty: (1) the average quarterly growth rate of revenue, and (2)variable cost as a percentage of revenue. We assume that average quarterly revenuegrowth is random and follows a normal distribution with mean = 5 percent, andstandard deviation = 5 percent. Variable cost as a percentage of revenue is also


FIGURE 5.12 Spreadsheet segment from Project.xls. Note that columns E through K are hidden.

normally distributed with a mean of 50 percent, and a standard deviation of 5percent. The discount rate is assumed to be 12.5 percent. Figure 5.12 shows aspreadsheet segment from Project.xls that is used to value this project with netpresent value (NPV). For the purposes of this example, we will use the Mean of theNPV forecast, ENPV, as the value of the project.

By looking at this project from different perspectives, we can estimate the valueof the project manager’s flexibility over time to affect the course of the project. Toestimate the value of this flexibility, we consider two scenarios to find:

ENPV 1 = The value of the project without managerial flexibility; andENPV 2 = The value of the project with managerial flexibility.

The value of the managerial flexibility is then determined as the difference betweenENPV 1 and ENPV 2.

With a standard NPV analysis, we assume that the project manager has noflexibility to make decisions during the life of the project. That is, once the project


is begun it is run to the end of three years with no expansion if successful, and noabandonment if unsuccessful.

Open the file Project.xls and run the simulation model. When the simulationstops, look at the Crystal Ball forecast window for NPV 1, which is also shown inFigure 5.13. The mean net present value is $18.61, which is the value of the projectwith no flexibility. Note that the output in Figure 5.13 also shows the probability isonly about 40 percent that the project will add value to the firm, because the valuein the certainty field at the bottom center of the forecast window is 42.92 percent.

Now suppose that we are faced with a similar situation, but this time we havethe option to abandon the project if unfavorable circumstances occur. For now,let’s use the following decision rule for abandonment. Begin checking in the secondquarter of Year 2, and abandon the project if three consecutive quarters of negativecash flow occur. This decision rule is built into rows 50 and 51 of Project.xls withindicator variables, as shown in Figure 5.14. Cells G50:M50 check whether or not toabandon each quarter, and cells G51:M51 ensure that once abandoned, the projectcontributes no positive cash flows in future quarters.

The Crystal Ball output in Figure 5.15 shows that the option to abandon hasincreased the project’s value. Because the mean NPV with abandonment possible is$50.62, the value of the abandonment option is estimated to be $50.62 − $18.61= $32.01. Note that the flexibility to abandon the project does not change theinherent risk structure of the project itself, so in Figure 5.15 the probability that theproject adds value to the firm remains the same, that is, Pr(NPV > 0) ≈ 40 percent.

FIGURE 5.13 Crystal Ball forecast for NPV1 for the case of no flexibilityin the Project.xls model.


FIGURE 5.14 Spreadsheet segment from Project.xls model showing the possibilityof abandonment. Note that columns E through K are hidden.

FIGURE 5.15 CB forecast window from Project.xls model showing thepossibility of abandonment.


However, by allowing for abandonment, we limit the losses incurred by the firm,just as an active and astute project manager would do in a real-world situation.Because of the ability to limit losses in the left tail of the distribution, the manager’sflexibility increases the ENPV of the project.

Now consider another option the manager has with this project, the option toexpand if favorable conditions occur. To value the expansion option, we use thefollowing decision rule. Begin checking in the second quarter of Year 2, and expandif we see three consecutive quarters of cash flow greater than $15. The investmentin the expansion project will cost $200, and we assume that expansion will doublethe quarterly revenue and expenses. This decision rule is built into rows 77 and 78of the spreadsheet model in Project.xls shown in Figure 5.16.

FIGURE 5.16 Spreadsheet segment from Project.xls model showing the possibilities of abandonmentand expansion. Note that columns E through K are hidden.


FIGURE 5.17 CB forecast window from Project.xls model showingthe possibility of expansion.

The Crystal Ball output in Figure 5.17 shows that the option to expand theproject has vastly increased its value. Because the mean NPV with both expansionand abandonment possible is $290.03, the value of the expansion option is $290.03− $50.62 = $239.41. Note that the flexibility to expand the project did change itsinherent risk structure slightly, in that the the probability that the project adds valueto the firm increased a bit, that is Pr(NPV > 0) ≈ 48 percent. If you extract thedata, you will see that on about 500 trials, the value of NPV 3 was positive whileNPV 2 was negative. In this situation we model the realistic behavior of an activedecision maker who will capitalize on fortuitous conditions that promote expansion.By expanding when times are good, the decision maker increases the ENPV of theproject by adding gains to the right tail of the distribution of possible results.

Threshold Values

In the previous example we arbitrarily specified the decision rule for project aban-donment as ‘‘abandon the project if three consecutive quarters of negative cash flowoccur.’’ Likewise, we arbitrarily specified the decision rule for project expansion as‘‘expand if we see three consecutive quarters of cash flow greater than $15.’’ Byadding decision variables to the model we can use the Decision Table tool to helpdetermine if the arbitrary threshold dollar amounts of $0 for abandonment and$15 for expansion are the optimal values to use. That is to say, it is possible thatsome other threshold values, such as −$5 and $20 might yield higher option valuesand if so we would like to find the threshold values that yield the highest optionvalues. Note that we could also use decision variables to determine whether using


FIGURE 5.18 Decision variables for the threshold value at which toabandon and expand in the Project.xls model.

two or four (or some other number) of consecutive quarters is optimal; however, forillustrative purposes here we will stick to threshold dollar amounts of cash flow.

To see the decision variable for the threshold value of cash flow for aban-doning the project, click on cell B4 in Project.xls. Then click on Define→DefineDecision. . . on the top menu in Excel. This will bring up the dialog window shownat the top of Figure 5.18. Cell B4 is named ‘‘Abandon Point,’’ and Crystal Ball willconsider four different values, {0, −3, −6, −9}. Note that we specified discrete stepsof $3, but by clicking the Continuous button in the Variable Type section of thedialog window, we could have had Crystal Ball investigate the potentially infinitenumber of values between virtually any specified lower and upper bounds. Using adiscrete variable type limits the number of values that Crystal Ball checks and thusspeeds up the analysis.


To see the decision variable for threshold value of cash flow for expanding theproject, click on cell B5 in Project.xls. Then click on Define→Define Decision. . . onthe top menu in Excel. This will bring up the dialog window shown at the bottomof Figure 5.18. We have named this decision variable ‘‘Expansion Point’’ and havetold Crystal Ball to consider seven different values, {0, 5, 10, . . . , 30}. Note that wespecified discrete steps of $5 here to speed up the analysis and simplify the resultsdisplayed in the next section.

Two-Way Decision Table

With the decision variables for cells B4 and B5 as defined in the previous section, wecan use Crystal Ball’s decision table tool to find the optimal combination of valuesfor Abandon Point and Expansion Point.

1. In the top menu, click on Run→Tools→Decision Table to bring up Step 1 ofthe Decision Table tool (see Figure 5.19). Select the target forecast NPV 3 asshown in Figure 5.19). Click the Next> button to continue.

FIGURE 5.19 Step 1 in using the Decision Table tool with the Project.xls model.



2. Use the >> button to move the decision variables Abandon Point and ExpansionPoint into the Chosen Decision Variables box on the right side of the dialog boxas shown in Figure 5.20. Click the Next> button to continue.

3. Take the default values for the fields in the step 3 dialog box as shown inFigure 5.21. Click the Start button to tell Crystal Ball to use the decision tabletool to evaluate all possible combinations of the specified values of AbandonPoint and Expansion Point.

You have just instructed Crystal Ball to run 10,000 trials for each of 4 × 7 = 28different combinations of decision variable values. The amount of time it takesCrystal Ball to run these 280,000 iterations depends on the speed of your computer.

Interpreting the Results

Figure 5.22 shows the results of using the decision table tool. Cell C3 containsthe maximum expected NPV of $299.78, which results from using the followingdecision rules: (1) Begin checking in the second quarter of year, and abandon theproject if three consecutive quarters of cash flow below −$6 occur; and (2) expand if



FIGURE 5.22 Results from using the Decision Table tool with theProject.xls model.


three consecutive quarters of cash flow greater than $5 occur. These new thresholdvalues of −6 and 5 are shown in the labels occupying cells C1 and A3, respectively,in Figure 5.22. Note that the maximum value of $299.78 identified in the DecisionTable tool output is $9.75 greater than the $290.03 identified in Figure 5.17. Keepin mind, though, that these figures are just estimates based on 10,000 runs of thesimulation. More runs would result in higher precision of the estimates.

At its heart, Monte Carlo simulation is just a computer-based sampling system.Thus, specifying more runs of the simulation will yield more precise results just asincluding more items in a random sample yields more precise estimates in a statisticalstudy designed to make inferences about a population. Crystal Ball allows you tospecify the level of precision that you desire. See Chapter 6 for information on howto specify the precision level.

In this example, we varied the Abandon Point in steps of $3, and the expansionPoint in steps of $5. We might be able to identify even higher levels of expected NPVby using smaller steps.

The Decision Table tool works well to find the optimal solution for problemsinvolving one or two decision variables. It also serves to introduce the notion ofsimulation optimization, which is the process of finding the best values of decisionvariables that we just completed for abandon point and expansion point. When morethan two decision variables are involved, as is typical for more realistic problems,the add-in OptQuest is a more powerful tool for finding an optimal solution. Thisis the topic of the next section.

USING OPTQUEST

We saw in the last section how a decision table could be set up to find the valueof a designated output for selected combinations of two decision variable values.Once that was accomplished, we found the best solution by looking at the valuesin the table to locate the combination that gave the maximum value of the output.Decision tables work very well for one or two decision variables. However, if thereare more than two decision variables, decision tables are cumbersome. This sectiondescribes how to use the Crystal Ball tool OptQuest to help find the best decisionsto make in situations involving more than two decisions.

Terminology

In order to understand better how OptQuest works, some background knowledge ofthe terminology is required. This subsection gives definitions for some of the termsthat will be used throughout the rest of this section.

Constraint. A constraint is a relationship among decision variables thatrestricts the values of the decision variables. When you define a decisionvariable, you constrain it individually by specifying the bounds. However,


TABLE 5.1 Forecast statistics available for optimizing or using in arequirement with OptQuest.

Mean Percentile Mean standard errorMedian Skewness CertaintyMode Kurtosis Final valueStandard deviation Coefficient of variabilityVariance Range

with an OptQuest constraint, you can restrict the values of linear combina-tions of decision variables. For example, an OptQuest constraint might beused to ensure that the sum of weights in a portfolio of assets is equal to1.0.

Objective. An objective gives a mathematical representation of the criterionby which it will be determined what is best or optimal. For the Project.xlsmodel, the objective is the maximization of mean net present value (ENPV)because NPV is a direct measure of the value added to the firm by actionsundertaken by a decision maker. However, other objectives could also bechosen, such as minimizing the cost or the riskiness of a decision.

Forecast statistic. A forecast statistic is a summary value of a forecast distribu-tion, such as the mean, standard deviation, or variance. The optimizationis controlled by maximizing, minimizing, or restricting a selected forecaststatistic. For example, in the Project.xls model, we chose to maximize themean NPV because doing so will, on average, give the best results. Thus,maximizing the mean of a forecast is a good strategy for a firm that makesmany decisions on a periodic basis. An individual or a firm that is morerisk-averse, however, might choose to minimize the standard deviation,or maximize the 5th percentile of the forecast distribution, for example.Table 5.1 lists the forecast statistics available for optimizing or using in arequirement.

Requirement. A requirement is a restriction on a forecast statistic that youcan set while trying to optimize on some other statistic. For example,you might be interested in maximizing the mean return on a retirementportfolio while at the same time requiring that the risk as measured bythe standard deviation does not exceed some specified value. You can userequirements to set upper and lower limits for any statistic of a forecastdistribution.

Example

Open Project.xls, then go to the top menu and select Run→OptQuest. In OptQuest,select File→New. This will start a wizard that will lead you through the dialogwindows to set up the tool. The Decision Variable Selection window appears first,


FIGURE 5.23 Decision Variable Selection window for the Project.xls model.

FIGURE 5.24 Constraints window for theProject.xls model.

listing every decision variable defined in the Crystal Ball model. Your window shouldlook like the one shown in Figure 5.23.

The window in Figure 5.23 lets you select which defined decision variables tooptimize. The columns and buttons are mostly self-explanatory, although it mightnot be immediately apparent that Suggested Value is simply the value that is in thedecision variable cell when OptQuest is started. To follow this example, you needonly to use the default values, so click OK to continue.

The next window that appears is the Constraints window, which is shown inFigure 5.24.

In OptQuest, constraints limit the possible solutions to a model in terms ofrelationships among the decision variables. In Version 2.3 of OptQuest, you can usethe Constraints window to specify only linear constraints. In the Project.xls model,we have no constraints on the decision variables (except for the bounds, which wespecify when defining the decision variables), so click on OK to continue.

After you exit the Constraints window, the Forecast Selection window appearsnext, listing all the forecasts defined in the model as shown in Figure 5.25. In theforecast row for NPV 3, click in the Select column. From the drop-down menu,select Maximize Objective, which by default will maximize the mean of the NPV 3


FIGURE 5.25 Forecast Selection window for the Project.xls model.

FIGURE 5.26 Options window for the Project.xls model.

forecast. Note that you can choose any of the forecast statistics in Table 5.1 tooptimize. Click on OK to continue.

The Options window lets you set options for controlling the optimizationprocess, as shown in Figure 5.26. See the OptQuest User Manual, pages 76--81, fora discussion of the choices available in this window. For now, we’ll accept all thedefaults, so click on OK to continue. Then click on Yes to indicate that you want torun the optimization now.

While the optimization is running, a Status and Solutions window and Perfor-mance Graph should appear, as shown in Figures 5.27 and 5.28, respectively.

When running an optimization, you can stop, pause, continue, or restart at anytime. You cannot work in Crystal Ball or Excel or make changes in OptQuest whenrunning an optimization, but you can work in other programs. Do not close Excel,Crystal Ball, or OptQuest while running an optimization.

After solving an optimization problem with OptQuest, you can

■ Run a solution analysis to determine the robustness of the results;


FIGURE 5.27 Status and Solutions window for the Project.xlsmodel.

FIGURE 5.28 Performance Graph for the Project.xls model.

■ Run a longer Crystal Ball simulation using the optimal values of the decisionvariables to assess with more precision the risks of the recommended solution;or

■ Use Crystal Ball’s analysis features to evaluate the optimal solution further.

For now, notice that the best solution identified in Figure 5.27 is the same asthat identified in Figure 5.22 using Decision Table, namely an Abandon Point of−6 and an Expansion Point of 5. For both Figures 5.27 and 5.22, we used four


different values for Abandon Point and seven different values for Expansion Point.If we used more values with Decision Table, the Decision Table results would takeup more cells than shown in Figure 5.22, and would also have taken longer togenerate. However, with OptQuest we could have easily specified an essentiallyinfinite number of values by specifying ‘‘continuous’’ with the pulldown menu inthe Type column of Figure 5.23. This would increase the time taken by OptQuestto search for an optimal solution, but it would result in values of Abandon Pointand Expansion Point that are closer to the overall optimal values. This is one reasonwhy OptQuest searches for optimality are usually preferred over Decision Tablesearches.

CHAPTER 6Selecting Run Preferences

N ow that we have covered the basics of setting up a Crystal Ball model and usingits forecasts to help you make decisions, we will take a closer look at the options

available to you through the Run Preferences menu to control the execution of yoursimulation models.

TRIALS

Figure 6.1 shows the Trials tab of the Run Preferences dialog. A Crystal Ball trial isthe process of generating random variates from the stochastic assumptions you havedefined for your model, evaluating the formulas that depend on these values, thencalculating and storing the forecast values.

When you click the Run button, Crystal Ball begins executing the steps depictedin Figure 6.2. The actions taken by Crystal Ball at each step are described as follows:

Start is where Crystal Ball prepares itself to run a simulation by looking for theassumption and forecast cells in your spreadsheet, and getting ready to storeforecast values in your computer’s memory.

Set Values is where each trial begins by Crystal Ball generating a random value foreach stochastic assumption and placing it in the corresponding assumptioncell.

Recalculate is where Crystal Ball instructs Excel to use the values that were justplaced in the assumption cells to update each cell in the spreadsheet thatdepends on the assumptions.

Get Results is where Crystal Ball takes the updated value from each of the forecastcells and stores it in memory. If you have enabled sensitivity analysis, CrystalBall also stores the current value of each assumption cell during this stepfor possible analysis later with sensitivity charts.

Stop marks the completion of each simulation trial. If none of the stopping criteriadescribed below have been satisfied, Crystal Ball returns to the Set Valuesstep as indicated in Figure 6.2.

End is where Crystal Ball returns control to Excel after one of the stopping criteria ismet, or after you have clicked on the Stop button in the Crystal Ball toolbar.

95


FIGURE 6.1 Run Preferences Trials tab.

Crystal Ball executes the simulation cycle automatically, but it also gives yousome control over selected aspects of the execution. This section describes thoseaspects.

Number of Trials to Run

This is the maximum number of trials you want Crystal Ball to run before it stops thesimulation. There are other stopping criteria, so sometimes the simulation will stopbefore the maximum number of trials is reached. In general, the more trials you run,the better (more precise) will be your solution in a statistical sense. A rule of thumbis to use no fewer than 2,000 trials (see Figure 2.7). With Crystal Ball’s ExtremeSpeed feature, 10,000 trials will execute quickly for moderately complex models, sothat is the number of trials specified in most of the examples shown in this book.

Stop on Calculation Errors

When this box is checked, Crystal Ball will stop the simulation when a numericalerror occurs in an Excel calculation. Numerical errors are often caused by dividingby some zero somewhere. For example, for arrays of values that have differentalgebraic signs, using Excel’s =IRR(values,guess) formula in a forecast cell cansometimes cause a numerical error.

Selecting Run Preferences 97

Start

Set Values

Recalculate

Get Results

Stop

End

FIGURE 6.2 Simulation cycle.

It is good practice always to have this box checked, as numerical errors areusually the result of errors in model logic and this feature of Crystal Ball will alertyou if any such errors are present. However, if you are using =IRR(values,guess)in your model and have difficulty getting Crystal Ball to run to completion, tryunchecking this box.

Stop When Precision Control Limits Are Reached

Checking this box enables Crystal Ball’s statistical precision-checking feature. Theprecision of an estimate is determined by the half-width of a (1 − α)100% confidenceinterval, which is computed by Crystal Ball for the mean as zα/2s/

√n, where

zα/2 is the (1 − α/2)100 percentile of the standard normal distribution, s is theStandard Deviation, and n is the number of trials from which the standard error iscomputed. As discussed in Chapter 2, the value of s/

√n is reported in the Statistics

View of the Forecast window as Mean Standard Error. Smaller values of s yieldsmaller half-widths, which yields more precise estimates. Each of the three terms inthe half-width calculation (zα/2, s, and n) can be used to increase the precision of theestimate of the mean.


The value of zα/2 is affected by the number you enter in the Confidence levelfield in Figure 6.1, which is your specification of 1 − α. For example, when youspecify a 95 percent confidence level, then zα/2 = z.975 = 1.96, which means that theprecision is measured as roughly two standard errors of the mean. While there maysometimes be good reason to change the value of 1 − α in particular situations,leaving the confidence level at 95% will suffice for most financial models.

Crystal Ball calculates the standard deviation of the forecast values, s, usingExpression 2.1 as the simulation progresses. While s cannot be affected by optionsspecified in the dialog shown in Figure 6.1, it is possible to increase precision byreducing s (or equivalently, the variance, s2) through so-called variance reductiontechniques, which require structural changes to the model and are described inAppendix C.

The precision of the estimate increases at a rate inversely proportional to thesquare root of the number of trials. During the simulation, Crystal Ball recalculatesthe half-width periodically, then stops the simulation when either the maximumnumber of trials or specified precision is reached, whichever comes first.

As an example of how to use the precision control feature, open the fileAccumulate.xls, and click on the Forecast cell D4. Then select Define→DefineForecast . . . to get the Define Forecast dialog shown in Figure 6.3. Select thePrecision tab. Note that you might have to click on the More icon—two arrowheadspointing down (

�

) in the upper right portion of the dialog to see the tabs. Checkthe two check boxes and punch the radio button indicated in Figure 6.3, and specify50000 as the absolute units of precision as shown. Click OK.

In Run Preferences, change the Number of trials to run to one million (enter1000000 with no commas), and make sure that Stop when precision control limitsare reached is checked. Then click OK. Press the Run button on the Crystal Balltoolbar. Your simulation model should stop running well before 1 million trialshave run, and you should see a Statistics View for Year 30 Wealth that looks likethat shown in Figure 6.4.

It is also possible to specify precision for the median, the standard deviation, ora selected percentile. This is done by following steps similar to those above for speci-fying the precision of the mean. See the Crystal Ball User Manual for specific details.

SAMPLING

Figure 6.5 shows the Sampling tab of the Run Preferences dialog, which has optionsrelated to the algorithms for generating the random numbers Crystal Ball uses todrive the simulation. This section gives a high-level overview of these options. For amore technical discussion, see Appendix B.

Random Number Generation

Random numbers are values between 0 and 1 that Crystal Ball generates to driveall the randomness in your models. The algorithm that Crystal Ball uses to generate


FIGURE 6.3 Define Forecast dialog showing the Precision tab.

FIGURE 6.4 Statistics View for Year 30 Wealth forecast showingthat a precision of $46,587 was reached in 40,000 trials.


FIGURE 6.5 Run Preferences Sampling tab.

these random numbers has the capability to produce over two billion differentvalues in such a way they appear to have the Uniform(0,1) distribution and be in acompletely ‘‘random’’ order, that is, the numbers appear to be serially independent.However, this random order is predetermined and is the same every time CrystalBall runs. See Appendix B for more details about the algorithm used by CrystalBall’s random number generator.

Think of these random numbers as being placed in the predetermined randomorder clockwise on the perimeter of a circle with a spinner arrow in the middle.Once the spinner arrow is at its starting point, Crystal Ball will select each randomnumber it needs sequentially in a clockwise direction from the circle beginning withthe number at the spinner’s starting point. With this image in mind, you can think ofthe Random number generation options on the Sampling tab as giving you controlover where to set the spinner to designate the starting point.

By checking the Use same sequence of random numbers check box and spec-ifying an Initial seed value as indicated in Figure 6.5, you are telling Crystal Ball toset the starting point for its random numbers at the same place each time you run asimulation. The Initial seed value must be an integer between 1 and 2147483647.The Use same sequence of random numbers option helps to compare results fordifferent models having the same stochastic assumptions. By using the same sequence


of random numbers in two models, any differences in performance are due todifferences in the models rather than sampling error.

If you do not specify an initial seed value, Crystal Ball will determine a seedfrom the number of milliseconds that have elapsed since Windows began runningon your computer. This of course will be different each time you run the simulation,so you can expect Crystal Ball to produce different forecast values on each run.

Sampling Method

Appendix B also provides some technical details and references on the differencesbetween Monte Carlo sampling (MCS) and Latin Hypercube sampling (LHS). Asthe statistical theory behind MCS is well known, it should be used for applicationswhere your intended audience is most critical, such as in academic research. LHS isa form of stratified sampling where all portions of the tails of a distribution are sureto be sampled. The statistical properties of LHS samples are the subject of currentacademic research, and no widely accepted methods of assessing precision with LHSare available yet. Nevertheless, LHS is probably the best choice for most practicalapplications. LHS uses slightly more computer memory than MCS, but this shouldnot pose a problem for most users.

SPEED

Figure 6.6 shows the dialog containing options that let you control how quicklyyour simulation model executes. Once it is built and debugged, you will ordinarilywant it to run as fast as possible. However, there are sometimes reasons for notdoing so, such as when you are demonstrating the model to a potential user of itsresults.

Run Mode

I suggest that you use Extreme Speed (ES) mode whenever you can, but you mustrealize that it imposes a few constraints on how you build your model because it usesvectorization to speed up the calculations. Essentially, this means that ES mode willperform hundreds of simulation trials at a time in RAM (random access memory),while normal speed performs them one at a time. Most of Excel’s 320 functionsare supported, but a few are not. The most commonly used unsupported functionis the string function. One common way for strings to be used is when an Excel IFcommand puts ‘‘Yes’’ or ‘‘No’’ in a cell that is referenced by another formula. ESmode will not work with the strings ‘‘Yes’’ or ‘‘No,’’ but works fine if you replacethese with 1 and 0, or the Excel Boolean values TRUE and FALSE. Consult theCrystal Ball User Manual Appendix C for a complete list of unsupported functionsand the Crystal Ball Web site for updates on this feature.


FIGURE 6.6 Run Preferences Speed tab.

Here are some tips for getting the most out of Extreme Speed:

■ Make sure that your computer has lots of RAM.■ Build your models so that the UsedRange (upper left rectangle of non-empty

cells) is minimized.■ Use Latin Hypercube sampling.■ Avoid references to other worksheets that are not required for your model.■ Avoid using large ranges in function arguments (for example, in the LOOKUP

function).

Normal speed should be checked when you use a function not supported by ESmode. Demo speed is useful for demonstrating to others how your model works andanimating Excel graphs.

Chart Windows

These two radio buttons let you specify the rate at which Crystal Ball redraws thechart windows. Because this takes time away from doing the calculations duringa simulation, suppress the chart windows to get the simulation completed mostquickly. This may be most useful when using OptQuest.


OPTIONS

Figure 6.7 shows the Run options, which are mostly self-explanatory. Whenyou are building and refining your model, you should always enableStore assumption values for sensitivity analysis, and do sensitivity analyses to helprefinement. Once you are satisfied with your model, you can disable storage to helpCrystal Ball run the model faster.

It turns out that having highly correlated assumptions can sometimes interferewith sensitivity analysis. For example, assume that you have a model for whichthe Forecast cell is highly sensitive to Assumption A, but not impacted at all byAssumption B. However, if Assumptions A and B are defined with a high (positiveor negative) correlation, the sensitivity analysis could indicate that the forecast issensitive to both Assumptions A and B just because B tends to follow A. Therefore,when using sensitivity analysis with correlated assumptions, you will want to disablethe correlations when you are doing sensitivity analysis. Just remember to turn itback on when you run the model for decision making.

The Run user-defined macros option is useful for power users of Crystal Balland Excel, but for most financial models Excel has built in all the functions thatyou will need. However, if you are facile with Visual Basic for Applications (VBA)

FIGURE 6.7 Run Preferences Options tab.


and wish to do things Excel doesn’t already do, or you wish to do extra processingduring each trial of the simulation, see the Crystal Ball User Manual for how to usemacros with a Crystal Ball simulation model.

Enabling Warn if insufficient memory will cause Crystal Ball to issue a warningdialog if you don’t have enough memory in your computer to run the simulation.The dialog lists several things you can do to help, but the best course of action is toput more RAM in your computer if you see this warning often.

STATISTICS

Figure 6.8 shows the options to change how percentiles are calculated and formatted.Punch the radio buttons to match your taste and that of the decision maker whowill use your model’s output. The Calculate capability metrics check box is usedprimarily for Six Sigma applications, so is not discussed further here.

FIGURE 6.8 Run Preferences Statistics tab.

CHAPTER 7Net Present Value and Internal Rate

of Return

N ow that we have completed your introduction to Crystal Ball, we will beginlooking at several different types of situations for which Crystal Ball models are

useful. We start with net present value (NPV) models, because using Monte Carlosimulation to develop distributions of NPV is a source of controversy among someacademics even though it is done routinely by practitioners. In this chapter, we willconsider both sides of the controversy and see some models where the distribution ofNPV can help the decision maker gain insight into the problem at hand. We will alsoconsider the pros and cons of using internal rate of return (IRR) as a Crystal Ballforecast. It is assumed that you are already familiar with these concepts. For morebackground information on NPV and IRR, see any introductory finance textbooksuch as Melicher and Norton (2006).

DETERMINISTIC NPV AND IRR

Suppose that you have the opportunity to purchase an annuity that costs you $100at Year 0, and is certain to return $30 to you at the end of each Year 1 through5. These cash flows are depicted in the Excel chart on the spreadsheet segment inFigure 7.1. Denote the cash flow at the end of Year t as Ct, and the relevant annualrate of interest as r. Then the NPV of the annuity is defined as

NPV =5∑

t=1

Ct

(1 + r)t+ C0. (7.1)

For the cash flows in Figure 7.1, if r = 10 percent then NPV = $13.72 as shownin cell B11. Therefore, the annuity is a good investment for any individual with arequired minimum rate of return of 10 percent because the investment’s NPV of$13.72 is greater than zero at that rate.

Be aware that the definition of NPV in Expression 7.1 is slightly different fromthat used by the Excel NPV function. To find the NPV of the annuity in Figure 7.1,

105


FIGURE 7.1 Spreadsheet segment to model annuity with deterministiccash flows of −$100 at the end of Year 0, and $30 at the end of Years 1through 5.

we use the Excel formula

=NPV(0.1,B5:B9)+B4, (7.2)

which is entered in cell B11 of NPV.xls. Most finance textbooks refer to thequantity calculated in this example by Excel’s NPV function as the present valueat end of Year 0 of the cash flows obtained at the ends at Years 1 through 5.To get the net present value, we also consider the investment (negative cash flow)

Net Present Value and Internal Rate of Return 107

FIGURE 7.2 Spreadsheet segment tomodel stochastic cash flows at the endof Years 1 through 5. Model 1 cashflows in Years 1 through 5 are IIDnormal(30,3). Model 2 cash flowsfollow an additive random walk withnormal(0,3) increments.

at Year 0, denoted by C0 in Expression 7.1. This can be confusing, but the NPVfunction has been defined this way for so many versions of Excel that Microsoft isunderstandably loath to change it at this point because so many of their existingcustomers are accustomed to the nontextbook definition and use it in many of theirexisting models.

As an alternative to NPV, we can also help decide whether to purchase the annu-ity by calculating its IRR. The IRR is defined to be the value of r in Expression 7.1that makes NPV = 0. Because there is no convenient closed-form expression forcalculating IRR, we use Excel’s IRR function to find it for us. Notice that there is con-sistency between the financial definition of IRR and Excel’s IRR function. Cell B12in Figure 7.1 shows that the IRR for the annuity is 15.24 percent. You can checkthis by replacing 0.1 with B12 in Formula 7.2 for cell B11 and seeing that NPV=0.

SIMULATING NPV AND IRR

Now let’s assume that we can purchase an investment product for $100 that hasstochastic cash flows in Years 1–5. We will use two different stochastic processesfor the risky cash flows, and compare the results to the annuity described in theprevious section.

Model 1. The cash flows at the end of Years 1–5 are independent and iden-tically distributed (IID) over time. Specifically, each cash flow is calculatedas Ct = 30 + 3Zt for t = 1, 2, 3, 4, and 5, where each Zt is drawn from a


normal(0,1) distribution independently of the Zts for the other years. TheModel 1 cash flows are in cells C5:C9 of file NPVModels.xls shown inFigure 7.2.

Model 2. The cash flows at the end of Years 1–5 are linked over timein an additive random walk model. Year 1 cash flow is computed asC1 = 30 + 3Z1, so is equal to Model 1’s Year 1 cash flow. Years 2–5 cashflows are computed as Ct = Ct−1 + 3Zt where the Zts for t = 2, 3, 4, and 5are the same normal(0,1) random variates used to generate Model 1’s cashflows. The Model 2 cash flows are in cells D5:D9 of file NPVModels.xlsshown in Figure 7.2.

To compare the effect of the IID model to that of the additive random walkmodel, look at the differences between distributions of NPVs and IRRs in Figure 7.3.Overlay Chart 1 in Figure 7.3 compares the distributions of NPVs for the two modelsof cash flow. Each distribution has the same true expected value, which is $13.72as it is for the annuity shown in Figure 7.1. However, a large difference in thevariability of the two distributions is evident in the overlay chart. As you can seein Figure 7.4, the standard deviation of the distribution of Model 1 NPV is $5.14,while in Figure 7.5 the standard deviation of the distribution of Model 2 NPV is$16.11. This difference in variability is explained by the difference in the modelsused to calculate the cash flows. Because the cash flows are linked to each other inthe random walk model (Model 2), their variability increases from year to year. Forexample, in Model 1 the cash flow for Year 5 is calculated as C5 = 30 + 3Z5, so hasa true variance of 32 = 9 and standard deviation of 3. In Model 2, the cash flow forYear 5 is linked to all previous years’ cash flows:

C5 = C4 + 3Z5

= C3 + 3Z4 + 3Z5

= C2 + 3Z3 + 3Z4 + 3Z5

= C1 + 3Z2 + 3Z3 + 3Z4 + 3Z5

= 30 + 3Z1 + 3Z2 + 3Z3 + 3Z4 + 3Z5,

so has a true variance of 5(32) = 45 and standard deviation 6.708. The increasingdispersion of cash flow distributions over time in Model 2 reflects the decisionmaker’s increased uncertainty about the cash flows the farther into the future he orshe looks. This increased uncertainty in cash flows causes the standard deviationfor Model 2 NPV to be larger than the standard deviation for Model 1 NPV inFigure 7.2. Overlay Chart 2 in Figure 7.3 shows similar differences in dispersion forthe distributions of IRR.

Using simulation to find a distribution of net present value is heresy to somefinance professors, yet many analysts do this routinely without knowing that it is con-troversial. When the concept of using distributions of NPV to compare investments


FIGURE 7.3 Overlay charts to compare distributions of NPV (Overlay Chart 1) and IRR (OverlayChart 2) for Models 1 and 2 in Figure 7.2. In both charts the distribution for Model 1 cash flows hasmuch less dispersion than the distribution for Model 2 cash flows.


was first promoted some 40 years ago by Hertz (1968), computers were not widelyavailable to managers as they are now. At that time, the only practical method ofcalculating present value available to financial analysts was to estimate the expectedvalue (mean) of potential cash flows for each future period and discount them aswe did for each deterministic Ct in Expression 7.1. Doing so ignores the variation inpotential future cash flows, and could lead the uninitiated to conclude that there is nodifference between the annuity and the investment with stochastic cash flows. How-ever, there is clearly a difference in the nature of these investments, and using simu-lation to help illustrate the differences can be eye-opening for many decision makers.

The controversy over whether to use simulation to calculate a distribution ofNPV stems from the definition of NPV long ago as the sum of discounted expectedcash flows. Under this definition, the NPV of any investment is a single number andsome adherents of this definition bristle at talk of a distribution of NPV. Proponentsof simulation, however, advocate finding the distribution of the sum of discountedpotential cash flows from an investment as we did in Figures 7.4 and 7.5, then usingthe distribution for analyzing the investment’s riskiness. When speaking to those whobristle, you may find it helpful to refer to a distribution such as those in Figure 7.4or 7.5 as a distribution of potential NPV rather than a distribution of NPV.

Notice that in this case the means of the distributions of potential net presentvalue are the same (within sampling error) as the NPV calculated as the sum ofdiscounted expected cash flows, $13.72. This will not hold true for all models, andin the next section we will see a model for which the sum of the discounted expectedcash flows will not be equal to the expected value of the distribution of the sum ofpotential cash flows, even after accounting for sampling error.

FIGURE 7.4 Split view of forecast chart and statistics window for Model 1 NPV in Figure 7.2.


FIGURE 7.5 Split view of forecast chart and statistics window for Model 2 NPV in Figure 7.2.

CAPITAL BUDGETING

For an illustration of using simulation for capital budgeting decisions, consider anexample from Chapter 10 of the excellent and popular finance textbook by Brealey,Meyers, and Allen (2006). I have tried to make the description below and the Excelmodels self-complete, but for a fuller discussion of this project see BMA 2006.

Figure 7.6 shows a spreadsheet model in ScooterNPV.xls for the Otobai Com-pany, who are considering the introduction of an electrically powered motor scooterfor city use. The inputs to be varied and their base-case values are in cells A4:B8.The five variable inputs are: market size, market share, unit price (yen), unit variablecost (yen), and fixed cost (billions yen). The model represented in cells A14:C24can be stated in the following expressions for Investment in Year 0 and the othervariables in Years 1–10:

Investment = 15 billion yen

Revenue = Market size × Market share × Unit price

Variable cost = Market size × Market share × Unit variable cost

Depreciation = Investment ÷ 10

Pretax profit = Revenue − Variable cost − Fixed cost − Depreciation

Tax = 0.5 × Pretax profit

Net profit = Pretax profit − Tax

Net cash flow = Net profit + Depreciation


FIGURE 7.6 ScooterNPV.xls spreadsheet model.

In the next section, we use this model to demonstrate the use of Crystal Ball’sTornado Chart tool for a deterministic sensitivity analysis of the project’s NPVto the model inputs. As is done in BMA 2006, the Tornado Chart tool considerschanges in each of the five inputs in turn while the other four are at their base-casevalues, and keeps track of the corresponding changes in NPV. Note that for eachof the five inputs we have defined a Triangular Assumption with parameters shownin cells C4:E8. This was done to facilitate use of the Tornado Chart tool, not toobtain a realistic simulation model for the investment. See Figure 7.14 for a realisticsimulation model that was created for risk analysis of this project.

Tornado Chart ToolTo use the Tornado Chart tool, select Run→Tools→Tornado Chart. . . from the topmenu. You will see a dialog window like that shown in Figure 7.7. Because NPVis the only Crystal Ball forecast defined in the spreadsheet, click Next > to indicatethat you wish to analyze the sensitivity of NPV to the inputs.

Figure 7.8 is the dialog window for Step 2. Click Add Assumptions to cause theinputs to appear in the list as shown in Figure 7.8. Note that we defined the inputsas Crystal Ball assumptions only to make this step easier. After the inputs appear inthe list, click Next >.


FIGURE 7.7 Step 1 in using the tornado chart tool.



The dialog for Step 3 of using the Tornado Chart tool is shown in Figure 7.9. Inthis window you can change some options that are self-explanatory. Note that if youtake the default values as shown in Figure 7.9, the tool will consider five differentlevels for each of the five inputs. The levels are determined by the percentiles of theCrystal Ball assumptions that were defined in cells B4:B8. Click Start to get theresults.

A tornado chart like that shown in Figure 7.10 will appear in a new Excelworkbook. This chart lists the inputs from top to bottom in decreasing order ofthe sensitivity of NPV to each input. Thus, Figure 7.10 indicates that NPV is mostsensitive to unit variable cost, and least sensitive to market size. The size of anybar corresponds to the magnitude of change in NPV. The color corresponds to thedirection of the change in NPV caused by an increase in the input. Because thebiggest bars are at the top and the smallest bars are at the bottom, the result is afigure that resembles a tornado. The tornado chart is useful for initial investigationof sensitivities to suggest the order in which we should be concerned with theinputs.

The spider chart in Figure 7.11 depicts the same information as the tornadochart. In this figure, it is the slope of any line that indicates the sensitivity of NPV to



FIGURE 7.10 Tornado chart. Darker bars indicate a positive change in the corresponding input.

changes in the corresponding input. For example, Figure 7.11 indicates that NPV ismost sensitive to Unit variable cost because the slope of that line is greatest. As theslope of the line is negative, we know that there is an inverse relationship betweenunit variable cost and NPV. The inputs appear in the same order in the spider chartas they do in the tornado chart.

The Tornado Chart tool is useful for a preliminary, deterministic sensitivityanalysis of any spreadsheet cell to its precedents. Although the tool did not requireus to do so, it made our job easier here to define NPV as a Crystal Ball forecast andthe inputs as Crystal Ball assumptions. When we do so, the output from the TornadoChart tool will be labelled with the names that we specified when we defined theforecast and assumptions.

A tornado chart resembles the chart provided by Crystal Ball’s SensitivityAnalysis feature, but it differs in how it obtains its results. Whereas the TornadoChart tool obtains results by varying the inputs one at a time deterministically, theSensitivity Analysis feature of Crystal Ball obtains its results through a statistical


FIGURE 7.11 Spider chart.

analysis of the relationships between the randomly generated inputs and outputscalculated during a set of simulation trials.

Risk AnalysisIn this section, we consider a simulation model of Otobai’s electric scooter projectthat follows the suggestions in Chapter 10 of Brealey, Meyers, and Allen (2006).Instead of assuming that the inputs are constant for Years 1–10 as we did above,we will allow them to vary by year. Furthermore, we will link them over time in amultiplicative random walk, and correlate the inputs for each year.

Figure 7.12 shows the five stochastic inputs and their base-case values incells A4:B8. However, we will introduce randomness through ‘‘forecast errors’’ foreach input denoted by ei,t for Inputs i = 1, 2, 3, 4, 5 and Years t = 1, 2, . . . , 10. Theindexing is as follows:

e1,t = Market size forecast error at year t

e2,t = Market share forecast error at year t


FIGURE 7.12 Part 1 of ScooterSimulation.xls.

e3,t = Unit price forecast error at year t

e4,t = Unit variable cost forecast error at year t

e5,t = Fixed cost forecast error at year t.

The simulation uses Crystal Ball’s Correlation tool to generate the forecast errorsfor each year as correlated normal random variates with mean zero and stan-dard deviations shown in cells E4:E8 in ScooterSimulation.xls. Figure 7.13 showsthe correlation matrix for the forecast errors generated in any given year. Theoff-diagonal elements on the upper triangular part of the matrix give the value ofthe correlation between errors during any given year. For example, the correlation

FIGURE 7.13 Correlation matrix for Crystal Ballassumptions defined for the forecast errors in columnscells C16:L20 of ScooterSimulation.xls.


between e1,t and e2,t is 0.8, while the correlation between e1,t and e4,t is −0.8.However, the errors are generated independently from year to year. These forecasterrors are in cells C16:L20, as shown in the spreadsheet segment in Figure 7.14.

The simulation obtains Year t = 1 values for each input by multiplying thebase-case value for each input by one plus its forecast error:

(Market size)1 = 1,000,000(1 + e1,1)

(Market share)1 = 10(1 + e2,1)

FIGURE 7.14 Part 2 of ScooterSimulation.xls. Note that columns Ethrough K are hidden.


(Unit price)1 = 375,000(1 + e3,1)

(Unit variable cost)1 = 300,000(1 + e4,1)

(Fixed cost)1 = 3,000(1 + e5,1)

Then the Years t = 2, 3, . . . , 10 values are obtained through the following set ofrecursive equations:

(Market size)t = (Market size)t−1(1 + e1,t)

(Market share)t = (Market share)t−1(1 + e2,t)

(Unit price)t = (Unit price)t−1(1 + e3,t)

(Unit variable cost)t = (Unit variable cost)t−1(1 + e4,t)

(Fixed cost)t = (Fixed cost)t−1(1 + e5,t).

This set of equations defines a multiplicative random walk for each input. Thus,with a combination of Crystal Ball’s Correlation feature and the linking of eachyear’s value to the previous year’s values, we are modeling interdependence betweeninputs and over time. For this example, we have specified some arbitrary values, butin practice a risk analyst will be able to estimate the parameters of the forecast errordistributions and the correlations from past projects using company data. Manyfirms have enterprise resource planning (ERP) systems from which historical datacan be retrieved to parameterize the simulation model.

Figure 7.15 shows the distribution of NPV for the scooter project. On average,the project is profitable, but the distribution shows that there is some risk involvedas there is nearly a 40% chance that NPV will be negative. Keep in mind that thesimulation model described here did not account for any managerial flexibility asdid the Project.xls model in Chapter 5. We can easily alter the BMA 2006 model tocontain a set of rules similar to those in the Project.xls model. If we build in rulessuch as those we used to model abandonment or expansion in Project.xls, we wouldsee a similar increase in NPV.

Figure 7.16 shows the distribution of IRR for the scooter project. Because weunchecked the box in Run Preferences for Stop on calculation errors as describedin Chapter 6, this figure can be somewhat misleading if you look only at the meanof 20%. In the upper left-hand corner we see that only 8,404 trials were completedeven though we specified a maximum of 10,000 trials and no other stopping criteria.A mixture of negative and positive cash flows can cause numerical problems withExcel’s IRR function, and such instances were simulated in the 1,596 trials notdisplayed because of error.

Finally, note that cells B12 and B13 in Figure 7.14 show values of $3.43 and15%, respectively, which differ by more than just sampling error from the means of$6.02 and 20% shown in Figures 7.15 and 7.16. This is a manifestation of Jensen’s


FIGURE 7.15 NPV Forecast for ScooterSimulation.xls.

FIGURE 7.16 IRR Forecast for ScooterSimulation.xls.


Inequality, which states that in general,

E[f (X)] �= f (E[X]). (7.3)

In words, this means that the values of NPV and IRR shown in Figure 7.14 (denotedby f (E[X]) in Expression 7.3) calculated by plugging in the mean values of thestochastic assumptions, are not equal to the expected values of NPV and IRRthat we are estimating in Figures 7.15 and 7.16 (denoted by E[f (X)] . Historically,academics have professed the use of f (E[X]) because of its easiness to calculate byhand. However, the ubiquity of Excel and the availability of Crystal Ball makescalculation of E[f (X)] just as easy nowadays.

Caveats

The use of simulation in capital budgeting has been controversial over the years, butas BMA 2006 point out, it can be a useful tool because the discipline of building amodel of a project can in itself lead you to a deeper understanding of the project.Once built and validated, experimenting with the model inputs will also further yourunderstanding.

An institutional advantage of building a simulation model is making explicit andsharing the assumptions that decision makers are using for planning. Oftentimes,different divisions of the same company will make different assumptions about aproject on which the divisions are collaborating. Heated discussions sometimes ensuebecause representatives from the different divisions are operating under different setsof assumptions for the same project, but they are not aware that their assumptionsets are different. Having a common simulation model of a project will help toensure that everyone is operating under the same assumptions.

CUSTOMER NET PRESENT VALUE

The general principle underlying customer lifetime value (CLV) analysis is thatcustomers are financial assets that organizations should manage just like any otherasset. Blattberg, Getz, and Thomas (2001), Reichheld (1996), Dwyer (1997), andothers have written much about the qualitative and quantitative aspects of managingcustomers to improve CLV. In this chapter, we will see a simple Crystal Ball modelthat demonstrates how to calculate distributions of CLV for customer segments withdifferent retention rates.

The model in Figure 7.17 is based on data given for the Buford Electronics casestudy presented in Blattberg, Getz, and Thomas (2001). In this example, there aretwo customer segments: the low-volume segment, who are customers from whomthe company receives revenues of less than $3,000 per year; and the high-volumesegment, who are customers from whom it receives revenues between $25,000 and$100,000 per year.


FIGURE 7.17 Model in LifetimeValueModel.xls for computing customerlifetime values for two different customer segments.

Cells C9:C18 and C22:C31 are Crystal Ball yes-no assumptions that take thevalue 1 if the customer quits buying from Buford during the year and 0 otherwise.We assume that Buford receives the revenues and pays the retention costs during theyear that a customer quits, but the customer never returns after quitting in any year.For each yes-no assumption, the probability of quitting is the complement of theretention rate in the same row of column B in the spreadsheet. Cells D10:D18 andD23:D31 use Excel =IF formulas to model the fact that once a customer quits, hedoes not return in a later year. The actual margins in cells H9:H18 and H22:H31 aregross profit margins earned from the customer if he is retained that year. These are


normally distributed random variables with means and standard deviations given incolumns F and G.

The values in cells I9:I18 and I22:I31 are the profits (Actual Margin minusRetention Cost) for each year, discounted to reflect the time value of money. Cells A4

FIGURE 7.18 Forecast charts for CLVs of two customer segments inLifetimeValueModel.xls.


and A5 use Excel’s =SUMPRODUCT formula to compute the NPV of each customersegment, which are defined as Crystal Ball forecasts.

Results

Figure 7.18 shows the results from 10,000 simulation trials. The low-volume cus-tomer segment is more profitable on average than the high-volume segment, and thedistribution of NPV for high-volume customers shows that a significant proportion(nearly 70 percent) are unprofitable, that is, have negative NPV. This differencein CLV is explained by the higher retention costs and lower retention rates of thehigh-volume customer segment. Some authors advocate a management approach toproactively identify and abandon unprofitable customers. See Blattberg, Getz, andThomas (2001) or Haenlein, Kaplan, and Schoder (2006) for more details aboutthis approach.

CHAPTER 8Modeling Financial Statements

P erhaps the most widely used financial models are the pro forma income statementand balance sheet. Most companies use deterministic versions of these models

for planning, and they are a natural place to start when constructing a CrystalBall model. Especially when decision makers are first exposed to risk analysis withCrystal Ball, it is best to make your stochastic models resemble as much as possiblethe deterministic models to which decision makers in your company are alreadyaccustomed.

In this chapter, we start with deterministic pro forma statements from Chapter 6of Sengupta (2004), which is an excellent source for more information on construct-ing deterministic financial models in Excel. In this chapter, we will focus on usingCrystal Ball with an existing deterministic model, just as you might do on the job.We walk through use of the basic tools to get you started. As you gain experience,you will be tempted to make your models far more complex than those presentedhere. However, do not add complexity just for its own sake. It is far better to startwith a simple model, and add complexity only if necessary to help make a sounddecision.

DETERMINISTIC MODEL

Figure 8.1 shows the historical income statement for 1999–2002 for the VitexCorporation example from Chapter 6 of Sengupta (2004). Figure 8.2 shows thebalance sheets for 1999–2002 for the Vitex Corporation. Our job is to project thesemeasures forward using historical data and input from management about the futureuncertainty.

We begin thinking about the uncertain future by looking to the past. While it isalways true that there are no guarantees the future will resemble the past, historicaldata are often the best information you will have available. Furthermore, if you dohave available to you a better source of information about what portends for yourcompany’s fortunes, then you can easily incorporate it in a Crystal Ball model usingthe methodology described here.

Figure 8.3 shows the common size statements for 1999–2002 for the VitexCorporation from Sengupta (2004). These are created by dividing each year’s

125


FIGURE 8.1 Historical income statement in Sengupta.xls.

income statement line items by the corresponding year’s sales so that each line itemin the common size statement is expressed as a percentage of sales. By looking at thevariability of common size line items over the years, we gain an idea of how muchvariability to expect in each line item. Then we can quantify the impact of each itemon the bottom line.

TORNADO CHART AND SENSITIVITY ANALYSIS

Figure 8.4 shows a Crystal Ball model for 2002–2006 for the Vitex Corpora-tion income statement from Sengupta (2004), which we use to create a tornadoand sensitivity chart. Cell J6 models the average changes in sales across years asa uniform(1%,10%) Crystal Ball assumption based on information provided bymanagement on the possible variation they see for sales. Cells J7, J10, and J12are uniform(48%,52%), uniform(27%,30%), and uniform(−1%,0%) assumptions,respectively. These parameters were selected based on the common size statements,and were defined to facilitate use of the tornado chart and sensitivity chart to seewhich variables had the greatest impact on the forecast, EBIT in 2006.

Figure 8.5 shows a tornado chart for the Crystal Ball model for 2002–2006 forthe Vitex Corporation income statement from Sengupta (2004). This shows that thesales forecasting factor had the largest impact, then cost of sales, followed by sales,general and administrative (SG&A) expenses.

Modeling Financial Statements 127

FIGURE 8.2 Balance sheet in Sengupta.xls.

FIGURE 8.3 Common-size statements in Sengupta.xls.


FIGURE 8.4 Crystal Ball model for income statement in Sengupta2.xls. Cells J6,J7, J10, and J12 are Crystal Ball assumptions, and I3 is a forecast.

FIGURE 8.5 Tornado chart for Crystal Ball model in Sengupta2.xls.The values of the Crystal Ball assumptions were varied from the 1stpercentile to the 99th percentile in five equally spaced steps.


FIGURE 8.6 Sensitivity analysis chart for Crystal Ball modelin Sengupta2.xls.

CRYSTAL BALL SENSITIVITY CHART

Figure 8.6 shows a sensitivity chart for the Crystal Ball model for 2002–2006 forthe Vitex Corporation income statement from Sengupta (2004). This chart showsresults similar to those obtained from the tornado chart, but indicates an even largerimpact from the sales forecasting factor. See Chapter 2 for an explanation of thedifferences between these two types of charts.

Figure 8.7 shows a Crystal Ball model for 2002–2006 for the Vitex Corpo-ration income statement from Sengupta (2004). Note that this models the annualchanges in sales and cost of sales as a percentage year-by-year as Crystal Ballassumptions. In Chapter 11, we look closer at modeling financial time series withthis sort of multiplicative random walk model, and introduce more models forsimulating financial time series.

Figure 8.8 shows a forecast chart for the Crystal Ball model for 2002–2006for the Vitex Corporation income statement from Sengupta (2004). A 95 percentcertainty interval for earnings before interest and taxes (EBIT) in 2006 is from$201.5 to $309.2.

CONCLUSION

In modeling financial statements, our intent is to project financial measures intothe future to help make informed decisions about the activities that result in these


FIGURE 8.7 Crystal Ball model in Sengupta3.xls.

FIGURE 8.8 Forecast chart for Crystal Ball model in Sengupta3.xls.

measures. Deterministic versions of pro forma statements have long been usedfor answering ‘‘what-if’’ questions about the bottom-line impact of managerialdecisions, and a Crystal Ball model can be viewed as a type of more-sophisticatedwhat-if analysis from which decision makers can gain more insight than from adeterministic model.


The procedure for modeling financial statements is to identify the key inputs, aswe did with the tornado chart, which varied many inputs one at a time to see theimpact on the key output, EBIT in 2006. Any output may be specified dependingon the decision maker’s interest. Because we used sales-driven forecasting, it isno surprise that we found sales growth to be the most important driver of 2006EBIT. This emphasizes the need to come up with a good model for defining salesassumptions.

We started with common-size statements to see which percentages are stableover time and which are variable. We usually base our projections on historicaldata where they are available, but can incorporate other educated guesses too, ifthey are encapsulated in probability distributions that are used to define Crystal Ballassumptions. When historical data are available, a popular technique is to obtainsubjective estimates from subject matter experts for the parameters of Uniform orTriangular assumptions.

The procedure for modeling financial statements is to identify the key inputs, aswe did with the Tornado Chart, then observe key outputs like EBIT in 2006. Anyoutput may be specified depending on the decision maker’s interest. Because we usedsales-driven forecasting, it is no surprise that we found sales growth to be the mostimportant driver of 2006 EBIT. This emphasizes the need to come up with a goodmodel for defining sales assumptions.

CHAPTER 9Portfolio Models

C rystal Ball is very useful for investigating different allocations of investment fundsto a set of risky assets. This chapter demonstrates the use of Crystal Ball and

OptQuest for determining the optimal allocation of funds in an investment portfoliobased on the decision maker’s risk tolerance. We use Crystal Ball and OptQuest tofind an optimal allocation for a situation where we know the true optimal allocation,and one where we do not.

SINGLE-PERIOD CRYSTAL BALL MODEL

In this example, we consider investing in the five asset classes listed in Table 9.1.Figure 9.1 shows a segment of the single-period Crystal Ball model in Portfolio.xls.Cells B12:E88 contain annual rates of return in percent on four asset classes. Theserates of return were calculated from the indices contained in the Indices worksheetof Portfolio.xls. The indices were constructed from data collected from varioussources for use only in the examples presented in this book. For more specific dataon asset returns available to investors during the period 1926–2002 (and more),see the Center for Research in Security Prices (www.crsp.com), Ibbotson Associates(2006), or Bodie, Kane and Marcus (2008).

Overview. Assume that you have four asset classes from which to choosefor an investment portfolio. These classes are listed in Table 9.1 along

TABLE 9.1 Means and standard deviations for annual total returns, 1 + ri, during theperiod 1926–2002 for four asset classes.

Mean Std.Asset Class Name Return Dev.

Large-Company Stocks LCS 1.1212 0.2052Small-Company Stocks SCS 1.1734 0.3607Corporate Bonds CB 1.0595 0.0794U.S. Government Bonds USGB 1.0559 0.0699

132

Portfolio Models 133

FIGURE 9.1 Spreadsheet segment from model to simulate a portfolio.

TABLE 9.2 Pearson correlation matrix for annual totalreturns during the period 1926–2002 for four asset classes.

LCS SCS CB USGB

LCS 1SCS 0.787 1CB 0.174 0.035 1USGB 0.104 −0.032 0.948 1

with their historical mean total returns, and standard deviations. ThePearson correlations for the five asset classes are in Table 9.2. The datafrom which these parameters were estimated are in Cells B12:E88 of theStochastic Model tab of Portfolio.xls.To keep things simple we will assume that you have $10,000 to invest(Cell A4) and wish to find the optimal percentage of your $10,000 to invest ineach of the asset classes (B8:E8). We ignore the effects of inflation for now.

Forecast. The forecast for this example is portfolio value in cell F8, the valueof the portfolio in Year 1.

Stochastic assumptions. The assumptions are defined in cells N13:Q13 byusing Batch Fit to find the distributions and Spearman correlations.


TABLE 9.3 Spearman correlation matrix for annual totalreturns during the period 1926–2002 for four assetclasses.

LCS SCS CB USGB

LCS 1SCS 0.811 1CB 0.124 0.047 1USGB 0.013 -0.045 0.936 1

The assumptions are referenced in cells B7:E7. Batch Fit was limitedto considering only normal or lognormal distributions for the historicalreturns. The normal distributions used for LCS and SCS were truncated atzero to reflect the limited liability of investing in equities. Batch Fit found thatlognormal distributions fit better to CB, and USGB. Lognormal distributionsare bounded by zero from below by definition, so needed no truncation. TheSpearman correlation matrix computed by Batch Fit is shown in Table 9.3.

Decision variables. Each decision variable in cells B8:E8 represents the per-cent of the initial investment allocated to the corresponding asset class. Eachdecision variable is defined with a lower bound of 0 percent, an upper boundof 100 percent, and a step size of 1.0 percent. By assuming a lower bound of0 percent we are precluding the possibility of selling short any of the assetclasses. The upper bound of 100 percent precludes borrowing to buy onmargin or selling short. The step size of 1 percent is specified to make the opti-mization converge on a solution more quickly than with a smaller step size.

Summary. The results shown here were found when using OptQuest tomaximize the mean of the total return forecast in cell F8, with the additionalrequirement that the standard deviation of total return be no more than$1,000. The optimal allocations are (27 percent, 11 percent, 0 percent, 62percent) for (LCS, SCS, CB, USGB) as shown in cells B8:E8 in Figure 9.1.For these allocations, the mean portfolio value is $10,865 with a standarddeviation of $998.87. By running OptQuest for longer than the 60 minutesused to obtain these results, one may be able to improve on the resultsslightly.

SINGLE-PERIOD ANALYTICAL SOLUTION

The worksheet Analytical Solution in Portfolio.xls shows the the optimal allocationto each asset class based on using Solver in Excel to maximize the mean returnsubject to the standard deviation of the portfolio remaining less than or equal to 10percent. The optimal allocation is (.22, .13, .00, .65) for (LCS, SCS, CB, USGB).This is the solution to the following mathematical programming problem:


maxα1,α2,α3,α4

E(P) =4∑

i=1

αiE(1 + ri)

subject to4∑

i=1

αi = 1

√αTSα = σ (P) ≤ 10%

0 ≤ ri ≤ 1 for all i,

where E(P) is the expected return on the portfolio, σ (P) is the standard deviation ofthe portfolio return, αi is the portfolio weight allocated to asset i = 1, 2, 3, 4 for theordering of assets (LCS, SCS, CB, USGB) as shown in cells J18:J21, and ri is themean rate of return for asset i. The 4 × 4 matrix S is the covariance matrix shownin cells H4:K7. For the optimal allocation, the expected return is 8.6 percent with astandard deviation of 10 percent.

It is not surprising, but is reassuring that this allocation agrees with the OptQuestallocation. For this simple problem, we can be certain that the deterministic solutiongives us the optimal allocation for the given values of the means, standard deviations,and correlations. Because OptQuest is a heuristic technique subject to samplingvariation, there is no guarantee that it will find the globally optimal solution.However, the fact that the OptQuest solution is so close to the known deterministicsolution in this simple case encourages us to believe that OptQuest will also findsolutions that are very close to the global optimum in problems that are toocomplicated for deterministic solutions to be used.

MULTIPERIOD CRYSTAL BALL MODEL

For investment advisors, a major consideration in planning for a client in retirementis the determination of an appropriate asset allocation that will enable the clientto withdraw funds necessary to maintain his or her desired standard of living. If aclient withdraws too much or if investment returns fall below expectations, there isa danger of either running out of funds or reducing the desired standard of living. Inthe model presented in this section we assume that the client is a woman. As womenhave slightly longer life expectancies than men, our results are conservative whenapplied to the retirement portfolio planning problem for a man of the same age.

The sustainable retirement withdrawal is the amount a client can withdrawperiodically from her retirement funds for a selected planning horizon. This amountcannot be determined with complete certainty because of the stochastic nature ofinvestment returns. In practice, the sustainable retirement withdrawal is determinedby limiting the probability of running out of funds to some specified level, such as


5 percent. The sustainable retirement withdrawal amount is typically expressed as apercentage of the initial value of the assets in the retirement portfolio, but is actuallythe inflation-adjusted monetary amount that the client will use each year for livingexpenses.

Suppose that at the end of 2002, a 60 year-old woman has $1 million in atax-deferred retirement account, and that she would like to withdraw $40,000 peryear in 2002 dollars. Assume that she has a life expectancy of 30 years, and that theinflation rate will be 3.12 percent. Her withdrawal in each year is $40,000 adjustedby the inflation rate. That is, her withdrawal at the end of 2003 will be $41,248, atthe end of 2004 will be $42,535, and so on.

In this scenario, her retirement withdrawal, or ‘‘spending rate’’ is specified at4 percent based on the initial balance of her total retirement funds. Her retirementportfolio planning problem is to allocate her initial $1 million to the asset classesavailable to her for investment, while maximizing her spending rate without runningout of funds before she dies. An optimal choice of spending rate and allocationscan be defined as one that limits to 5 percent the chance that she spends all ofher accumulated wealth at the end of a deterministic, 30-year planning horizon. Asa secondary issue, she may also be concerned with the value of her estate that isbequeathed to her heirs when she dies.

For this model, the data in Portfolio.xls were used to parameterize the CrystalBall model by using Crystal Ball’s Batch Fit tool. Table 9.1 shows the means andstandard deviations of the annual returns (in percent) during the period 1926–2002for four asset classes: large-company stocks (LCS), small-company stocks (SCS),corporate bonds (CB), and U.S. government bonds (USGB). Table 9.3 shows theSpearman correlation matrix for these four asset classes during the same period. Theannual rate of inflation during 1926–2002 averaged 3.12 percent.

The model in SustainableRetirementWithdrawals.xls generates stochasticreturns in cells I11:L71 for the assets LCS, SCS, CB, and USGB in years 2003–2063.The four returns in each row are correlated with each other, but each row is statisti-cally independent of the other rows. The Spearman correlations, in cells G78:J81,were computed by the Batch Fit tool. Crystal Ball’s Correlation Matrix tool was usedto create the upper triangular matrix in cells K78:N81, which references the Spear-man correlations. These are the values used by Crystal Ball during the simulationtrials.

The portfolio weights of the asset classes in cells I8:L8 are defined as decisionvariables in the range [0, 1] (i.e., no short sales nor margin purchases are allowed)in steps of 1 percent. At the end of each simulated year, a constant real amount (in2002 dollars) is withdrawn for living expenses. The withdrawal amount is definedas both a decision variable and a forecast variable in Crystal Ball. The portfolio isassumed to be composed entirely of tax-deferred dollars and the effects of taxes onthe amounts withdrawn are not considered.

We consider two different planning horizons: (1) a deterministic 30-year hori-zon, and (2) a stochastic horizon equal to the remaining lifetime of the woman


FIGURE 9.2 Custom distribution representing the death age of a 60-year-old female as given by the2001 CSO mortality table.

characterized by the 2001 Commisioner’s Standard Ordinary (CSO) mortality table.The distribution of the death age of a 60 year-old woman is shown in Figure 9.2.

In OptQuest, the percentage allocation to each asset class and the withdrawalrate are specified as decision variables. The percentage allocations are bounded by 0and 100 and are constrained to sum to 100 percent. An indicator variable is definedas a Crystal Ball forecast for the event that wealth at the end of each planninghorizon is positive. The withdrawal rate forecast variable in cell H4 is specified asthe objective to be maximized with an additional requirement that the mean of thepositive-wealth indicator variable have a lower bound of 0.9540. To account forsampling error in the estimates used by OptQuest in its optimization algorithm, thislower bound exceeds 0.9500 by approximately two standard errors of the mean ofthe positive-wealth indicator variable resulting from 4,000 trials. Table 9.4 showsthe allocations obtained for two different planning horizons:

1. A deterministic horizon of 30 years.2. A stochastic horizon equal to the woman’s remaining lifetime.

From a final run of 10,000 trials of the Crystal Ball model for the deterministic,30-year horizon, the estimated median value of the woman’s wealth is $3.79 million,with a 95.30 percent probability of being solvent (wealth greater than zero) at theend of 30 years. Figure 9.3 shows the distribution of wealth at the end of 30 years.For the stochastic, remaining lifetime horizon, the estimated median value of her


TABLE 9.4 Optimal asset allocations and sustainablewithdrawal rates for two planning horizons.

Horizon LCS SCS CB USGB Spend

30-year .25 .12 0 .63 4.08%Rem. life .23 .13 0 .64 4.53%

estate is $1.78 million, with a 95.25 percent probability of leaving to her heirs anestate greater than zero. Figure 9.4 shows the distribution of her estate.

Her spending rate is 4.08 percent with a fixed 30-year horizon, and 4.53 percentwith a stochastic horizon. With an initial investment of $1 million, the differencebetween the spending rates amounts to an additional $4,500 in 2002 dollars tospend each year. This differential stems primarily from the fact that the retiree hasa high likelihood of dying before she reaches age 90. For the assumptions statedabove, our analysis quantifies the risk of dying broke if one chooses to withdrawmore to live better during retirement.

From comparison of the results in Table 9.4 and Figures 9.3 and 9.4, it is evidentthat very large stock allocations are not necessary for sustainability of withdrawals.In fact, allocation of a majority of the portfolio to equities will increase the likelihoodof depleting the retiree’s funds during her lifetime. For both planning horizons, thesplit between debt and equity in the retirement portfolio is roughly 60–40.

FIGURE 9.3 Distribution of wealth after 30 years with asset allocationslisted in Table 9.4 for the 30-year planning horizon.


FIGURE 9.4 Distribution of the retiree’s estate with asset allocations andwithdrawal rate listed in Table 9.3 for the remaining-lifetime planninghorizon.

This example is intended only to demonstrate the use of Crystal Ball and Opt-Quest for financial planning. A more thorough analysis would include analyses ofother potential investments, such as real estate or international equities, and morespecific and recent data on the components of the asset classes used in this chapter.However, the analysis presented here does serve to inform individuals who are facingretirement about the tradeoffs involved in the retirement portfolio planning problem,and gives financial planners an idea of how to use Crystal Ball and OptQuest todemonstrate to their clients the risks involved in retirement planning.

CHAPTER 10Value at Risk

M any times one wants to know for planning purposes what is the ‘‘worst that canhappen.’’ In many situations, the worst that can happen is to lose one’s entire

investment; however, this usually has an extremely low probability of occurrence.The concept of Value at Risk (VaR) was devised to obtain a risk measure thatassociates a severe loss with a probability level of reasonable interest to the decisionmaker, such as 1 percent or 5 percent. See Jorion (2001) for more about VaR. Inthis chapter, we see how to use Crystal Ball to find VaR and a related measure,Conditional VaR (CVaR).

VAR

In practice, we can think of a potential loss L as the worst that can happen if theprobability of losing L or more during a selected time period is a specified amountsuch as 5 percent. In that case, L is called the ‘‘5 percent Value at Risk (VaR).’’More precisely, let R denote the total return (in dollars) on an investment, I, and letc denote the α percentile of the distribution of R. Then the α percent VaR is definedas L = I − c.

Figure 10.1 shows a segment of the one-year Crystal Ball model in PortfolioVaR.xls, which is adapted from the file Portfolio.xls described in Chapter 9. The potentialloss from investing in the portfolio, I − R, is measured directly in cell B11 with theExcel formula =A4-A11, which is simply the difference between the initial investmentand the final value of the portfolio. A copy of the forecast window for this quantity isshown at the bottom of the spreadsheet segment in Figure 10.1. Because the certaintyis 95 percent that portfolio loss is between −Infinity and $792.47, we say that the 5percent VaR for one year is $792.47. Note that when we find the α percent VaR fromthe loss (I − R) distribution, we use the (1 − α) percentile in the upper tail rather thanthe α percentile, c, in the lower tail of the distribution of R.

VaR is used by regulators to compute capital requirements for financial insti-tutions, and by managers as an input to risk-management decisions. VaR can alsobe used by managers to assess the quality of their models. For example, if a modelprovides that there is a 5 percent chance that a bank’s trading operations will lose$1 million over a 1-day horizon, then on average the trading operation should lose

140

Value at Risk 141

FIGURE 10.1 Spreadsheet segment from model to illustrate the concept of Value at Risk (VaR).

$1 million or more on 5 percent of the days in a randomly selected period. If thereare many more losses, it implies that the model assigns too little risk to the situation.If there are many fewer losses, it implies that the model assigns too much risk. Inthis sense, VaR can therefore be used to check the validity of a model.

Some regulators have adopted VaR as part of their risk management guidelines,but critics of this measure have pointed out some of its shortcomings, which arediscussed in the next section.


SHORTCOMINGS OF VAR

VaR provides no information about the extent of losses that might occur beyondthe threshold level. In that sense it is very optimistic because it gives a lower boundon potential loss at the α percent level. Further, it is not always subadditive, whichmeans that the VaR of the combination of two or more investments can exceedthe sum of the individual VaR for each investment. This is contrary to the basicprinciple of diversification, which holds that risk will decrease when more assets areheld, not increase. This failure to reward diversification is perceived as the greatestshortcoming of VaR. An alternate risk measure, Conditional Value at Risk (CVaR),overcomes these shortcomings.

CVAR

Conditional Value at Risk (CVaR) is the expected value of losses beyond thethreshold level. Figure 10.2 shows a forecast window for the portfolio loss inPortfolioVaR.xls. During the definition of cell C11 as a forecast, a filter was set onthe forecast values to exclude values in the range −Infinity up to $792.47. That iswhy ‘‘500 Trials’’ appears in the upper left part of the forecast window, even though10,000 trials were run. The mean of these 500 largest forecast values is $1,192.56.In general, the α percent CVaR is the expected value of losses that exceed the α

percent VaR level. Figure 10.3 depicts the 5 percent VaR and CVaR on the lossdistribution for a one-year holding period for the portfolio in PortfolioVaR.xls.

FIGURE 10.2 Filtered forecast window for Portfolio Loss inFigure 10.1.

Value at Risk 143

FIGURE 10.3 Forecast window for Portfolio Loss in Figure 10.1showing the 5 percent VaR over one year of $1,967 and thecorresponding CVaR of $2,794.

Investments with high CVaR will necessarily have high VaR as well. CVaR,which is also called Conditional Tail Expectation, Expected Tail Loss, Mean ExcessLoss, Mean Shortfall, or Tail VaR, is considered to be a more ‘‘coherent’’ measureof risk than VaR. Artzner, Delbaen, Eber, and Heath (1999) describe subadditivityand other coherent measures of risk in detail. Uryasev (2000) and Hardy (2006) aregood references on the basics of CVaR.

CVaR.xlsThe file CVaR.xls, a segment of which is shown in Figure 10.4, contains a simulationmodel of a three-asset portfolio. Assets 1, 2, and 3 have mean returns of 10 percent,12 percent, and 13 percent, respectively, with variance-covariance matrix

0.10 0.04 0.030.04 0.20 −0.040.03 −0.04 0.30

.

The variance-covariance matrix is contained in cells B9:D11, and the correspondingPearson correlation matrix is computed in cells B14:D16. As correlation matricesare symmetric, it appears as a lower triangular matrix. The variance-covariancematrix is used in an Excel array formula in this example (cell C32), so the entiresymmetric matrix is in the file.

The Crystal Ball model simulates returns from a portfolio with weights 0.30,0.25, and 0.45, invested in Assets 1, 2, and 3, respectively. Asset rates of return are


FIGURE 10.4 Crystal Ball model to find VaR and CVaR.

normally distributed, but truncated at −100 percent. The model has four forecastcells: Asset1 Value, Asset2 Value, Asset3 Value, Total Value. To reproduce theresults tabulated below, run 10,000 Trials with LHS and Seed = 813.

To get the 1 percent VaR and CVaR values from Crystal Ball, first find the 1stpercentile of each forecast using using the command

=CB.GetForePercentFN(Range,Percent)

as shown in cells B25:E25. Then use

Preferences→Forecast . . . →Filter Tab

to set a filter on the forecast values to include values in the range −Infinity to the 1stpercentile (entered as a number) of each forecast. The means of the filtered values areused to compute the α = 1 percent CVaR for each asset and the portfolio as shownin Table 10.1. Again, CVaR is preferred by some analysts because it is subadditive,which means that the CVaR of the portfolio is always less than or equal to the sum

Value at Risk 145

TABLE 10.1 VaR and CVaR for three assets and portfolio modeled inCVaR.xls.

1st Percentile Investment VaR CVaR

Asset1 Value $ 10.98 $ 30.00 $ 19.02 $ 21.86Asset2 Value $ 4.03 $ 25.00 $ 20.97 $ 22.72Asset3 Value $ 4.20 $ 45.00 $ 40.80 $ 42.76Portfolio Value $ 49.05 $ 100.00 $ 50.95 $ 56.75

of the individual CVaRs of the portfolio components. VaR is not always subadditive(although it is so in this example), which means that the risk of a portfolio can belarger than the sum of the stand-alone risks of its components when measured byVaR. The CVaR is also less sensitive to changes in the defining percentile α thanis VaR.

CVaRSubadditivity.xls

The model in Figure 10.5 is a Crystal Ball simulation of an analytical result presentedby Tasche (2002). The model simulates investments in each of two independent

FIGURE 10.5 Crystal Ball model to demonstratesubadditivity of CVaR in a situation for which VaR is notsubadditive.


TABLE 10.2 VaR and CVaR forindependent investments modeled inCVaRSubadditivity.xls.

1% VaR 1% CVaR

Total Loss $ 209.51 $ 4,687Loss1 $ 98.22 $ 2,392Loss2 $ 99.27 $ 2,407

opportunities with potential losses X1 and X2. Both X1 and X2 loss distributionshave Pareto distributions with Location Parameter = 1 and Shape Parameter =1. Cell D4 is a Crystal Ball forecast representing the loss on the first opportunity,Loss1 = X1. Cell D5 is a Crystal Ball forecast representing the loss on the secondopportunity, Loss2 = X2. Cell D6 is a Crystal Ball forecast representing the totalloss, Loss1 + Loss2. The VaR and CVaR of the Loss1, Loss2, and Total Loss areshown in Table 10.2. The table shows that CVaR is subadditive because the CVaRfor Total Loss, $4687 is less than the sum of the CVaRs for Loss1 and Loss2,$4799 = $2392 + $2407. However, VaR is not subadditive because the VaR forthe Total Loss, $209.51, is greater than the sum of the VaRs for Loss1 and Loss2,$197.49 = $98.22 + $99.27.

In other words, the 1 percent VaR for the portfolio is greater than the sumof the 1 percent VaRs of the investments considered individually. The principleof diversification holds that risk is lower when two or more assets are combinedinto a portfolio. As VaR indicates in this example that the risk of holding theportfolio is greater than the sum of the risks of holding each portfolio componentindividually, VaR is not a satisfactory measure of risk in this example. However,CVaR indicates correctly that the risk of holding the portfolio is lower than the sumof the component risks, and so it is a satisfactory measure of risk from this point ofview.

CHAPTER 11Simulating Financial Time Series

I n financial modeling, we encounter two main types of time-series data:

1. Observations that appear to be independent and identically distributed (IID).2. Observations that do not appear to be IID because they follow a trend or some

other pattern over time.

Financial theory provides a compelling argument—the efficient markets hypothe-sis—that returns on investments must be independent over time because no onehas access to information not already available to someone else. If returns areindependent, however, prices will be dependent over time and we will require a wayto model that dependence. This chapter presents some models that can be used forprojecting future returns, asset prices, and other financial times series in simulationmodels for risk analysis.

WHITE NOISE

A white noise process is defined to be one that generates data appearing to be IID.It takes its name from the fact that no specific frequency or pattern dominates ina spectral analysis of the observations, similar to white light, or the noise of staticemitted from an AM radio that is not tuned in to a station.

The model for a white noise process is

Wt = µ + εt, (11.1)

where µ is a constant, and εt is a sequence of uncorrelated random variablesidentically distributed with mean zero and finite variance for t = 1, . . . , T. Theprobability distribution of εt is not necessarily normal, but if it is the process is saidto be Gaussian white noise named after the eighteenth-century mathematician, CarlF. Gauss, who studied the properties of the normal distribution.

For example, we can simulate observations from a Gaussian white noise processwith Crystal Ball by placing several uncorrelated Normal(0,10) assumptions in acolumn, adding a constant, say µ = 200, and plotting the results as was done in thefile RandomWalk.xls. Figure 11.1 shows the model. In cells B6:B35 are Crystal Ball

147


FIGURE 11.1 Model to compare a white noise process to arandom walk. Note that rows 8 through 33 are hidden.

normal(0,10) assumptions that we denote as εt for t = 1, . . . , 30. A Gaussian whitenoise process was generated in cells C6:C35 using Expression 11.1, and a time seriesplot of one realization of the process appears in the center of Figure 11.1. Noticehow the independence of the observations in the white noise process is manifestedin the choppiness of its plot. For the white noise process, no matter where each

Simulating Financial Time Series 149

observation falls, the next observation is equally likely to be above or below themean of 200. This characteristic causes the choppy look.

RANDOM WALK

One form of a non-IID process is the additive random walk process defined by

Yt = Yt−1 + εt (11.2)

for t = 1, . . . , T. For example, in RandomWalk.xls, we set T = 30 and Y0 = 200,then generated observations from the process in Expression 11.2 in cells D6:D35,using the values in B6:B35 for εt, t = 1, . . . , 30.

A time series plot of one realization of the random walk process appears in thelower time series plot in Figure 11.1. Notice how the random walk process exhibitsa meandering pattern. The first few points are below the mean, then once the plotgoes above the mean, it tends to stay above for a while, then heads down and goesbelow the mean again before eventually heading back up. Even though the changesin the level of the random walk are independent, the levels themselves are dependentover time. This dependence causes more variability in the levels of the random walkprocess than is evident in the levels of the white noise process.

The aggregate effect of the dependence of the levels of the random walkcompared to the white noise process can be seen in Figures 11.2 and 11.3. Becausethe observations in the white noise process are IID, the forecast chart in Figure 11.2has a mean of µ = 200 and standard deviation of σ = 10, as do all of the observations

FIGURE 11.2 Forecast chart for the observation at time t = 30 forthe random process.


FIGURE 11.3 Forecast chart for the observation at time t = 30 forthe random walk.

Wt in the white noise process for t ≥ 1. The mean of the additive random walkprocess is also 200 for every observation, but the standard deviation grows largerevery time period because we are adding on another random change. It can beshown that each value Yt of the random walk process has a mean of µ = 200 anda standard deviation of σ

√t. In Figure 11.3 you can see that the standard deviation

(10√

30 = 54.77) is much greater than the standard deviation (10) of the forecastin Figure 11.2. The scales of the horizontal axes of these two plots were specifiedto be equal so that the difference in variability between the white noise process andrandom walk was apparent. However, the scales of the vertical axes in Figures 11.2and 11.3 are different. Figure 11.4 is another illustration of the differences in theseforecasts with an overlay chart for cells C35 and D35.

For a dynamic illustration of the difference between white noise and a randomwalk, see the file RandomWalk.xls. In Run Preferences, set Run Mode to Demoand watch the time series plots to see the difference in behavior when the simulationis running. The white noise process will bounce almost entirely within the 3σ boundsof 170 to 230 at every point in time, while the random walk will exhibit increasingvariability as t gets larger.

AUTOCORRELATION

Chapter 4 showed how to calculate both Pearson and Spearman correlations betweentwo variables with Excel. When checking for independence of a series of values overtime, we calculate the autocorrelation, which is the correlation coefficient of thevalues in the series that are separated by a specific length of time. In this context,


FIGURE 11.4 Overlay chart to compare the time t = 30 observationsfrom a white noise process and a random walk.

the prefix auto–means same, so the autocorrelation is the correlation of the valuesin a time series with other values within the same series. Sometimes authors refer toautocorrelation by the term serial correlation to emphasize the correlation within atime series.

While the correlation coefficients for values separated by two or more timeperiods are also of interest in time series analysis, for our purposes it is sufficientto think only about first-order autocorrelation, which is the correlation betweenvalues in a time series that are separated by one unit of time. Thus, first-orderautocorrelation is also called Lag-1 autocorrelation. Unless specified otherwise, theterm autocorrelation in this chapter is meant to refer to first-order autocorrelation.It is usually true with financial time series that if the first-order autocorrelation isnear zero, then the rest of the autocorrelation coefficients will also be near zero.However, for time series that exhibit seasonality, higher-order autocorrelation couldbe significant while lag-1 autocorrelation is low.

To calculate the first-order autocorrelation coefficient for the white noise processvalues in cells C5:C35, we entered into cell C3 the Excel formula =CORREL(C5:C34,C6:C35). As shown in Chapter 4, this calculates the Pearson correlation for thetwo arrays C5:C34 and C6:C35. Likewise, cell D3 holds the Excel formula=CORREL(D5:D34,D6:D35) to find the first-order autocorrelation coefficient forthe random walk time series in cells D6:D35. Note that there are other methodsto calculate the autocorrelation coefficient having more appeal to purists, but Exceldoes not yet include these other methods in its arsenal of statistical functions. Formore discussion of this point and other methods for calculating autocorrelationcoefficients, see pages 330–340 of Priestley (1981), or section 2.2 of Tsay (2002).


Of course, with more work, you can always use Excel to calculate the autocorre-lation coefficient by one of the other methods. For example, another way to calculatethe first-order autocorrelation coefficient, ρ1, for observed values yt, t = 1, 2, . . . , T is

ρ1 =∑T

t=2(yt − y)(yt−1 − y)∑T

t=1(yt − y)2,

where y = ∑Tt=1 yt/T. This version of the autocorrelation was calculated in cell Q6

for the white noise process in cells C6:C35 of RandomWalk.xls.To see how these autocorrelation coefficients vary during simulation trials, they

have been defined as Crystal Ball forecasts. Figure 11.5 shows the forecast chart forcell C3, the autocorrelation coefficient for the white noise process values. By the waythese values were generated, we know that they are independent over time, so theirtrue autocorrelation is zero. However, in any given simulation trial the calculated(sample) autocorrelation coefficient can differ from zero simply because of samplingerror. It can be shown that the sampling error for the first-order autocorrelationcoefficient calculated for an IID time series of length T has a standard deviationof approximately 1/

√T, so we would expect the standard deviation of the 10,000

values plotted in Figure 11.5 to be 1/√

30 = 0.183 and roughly 95 percent of thevalues to fall within the two standard error interval (−0.366, 0.366). Figure 11.5shows that 95.86 percent of the observations actually fell within that interval duringthe 10,000 simulation trials, which agrees with what we expect. Furthermore, the

FIGURE 11.5 Forecast chart for the autocorrelation coefficient for therandom process.


FIGURE 11.6 Forecast chart for the first-order autocorrelationcoefficient for the random walk process.

sample standard deviation of the distribution in Figure 11.5 is 0.178, which is alsoclose to its expected value of 0.183.

Figure 11.6 shows the autocorrelation for cell D3, the autocorrelation coefficientfor the random walk time series. All values of the random walk autocorrelationcoefficient were significantly larger than zero, which is what we expect because thelevels of the random walk process are not independent over time.

To check for a white noise process in practice, you can use the following teststatistic. First, calculate the first-order autocorrelation coefficient, ρ1, from the Ttime-series observations. Then find Z = ρ1

√T. If the absolute value of Z is greater

than two (|Z| > 2), conclude that the observations do not come from a white noiseprocess. If you reach this conclusion, then you must decide how best to model thetime series if you want to use Crystal Ball to generate potential future values of thetime series. The rest of this chapter describes some models for you to consider. Thereare many models that might be applied, but we show a few of the more popularmodels for generating future values of financial times series with Crystal Ball.

Selected models for simulating financial time series are popular because of some‘‘stylized facts’’ recognized by finance practitioners, and listed in McNeil, Frey, andEmbrechts (2005). For series of daily returns, exchange rates, and commodity prices:

■ Return series are not IID although they show little serial correlation.■ Conditional expected returns are close to zero.■ Volatility appears to vary over time.■ Return series are leptokurtic or heavy-tailed.■ Extreme returns appear in clusters.


Varying volatility and clustered, leptokurtic returns can be modeled with some typeof mixture model. The remainder of this chapter describes some models that can beincorporated in risk analysis spreadsheet models.

ADDITIVE RANDOM WALK WITH DRIFT

The model for an additive random walk with drift is

Yt = µ + Yt−1 + εt (11.3)

for t = 1, . . . , T, where µ is the mean change per time period and εt is an IIDsequence of random variables that is not necessarily normally distributed.

By subtracting Yt−1 from both sides of Expression 11.3, we get

Yt − Yt−1 = µ + εt,

which means that changes in the levels of a random walk with drift process followa white noise process.

Generating Values from a Scalar Random Walk with DriftProcessTo simulate potential future values of a time series that you think follows an additiverandom walk with drift process, take the first differences of the time series and fit aCrystal Ball assumption to them. This is illustrated in Figure 11.7.

The values of the time series Yt for t = 1, 2, . . . , 20 in cells B4:B23 ofRandomWalkWithDrift.xls are quarterly sales of an industrial product. The firstdifferences are found by entering =B5-B4 in Cell C5 and copying this formula downthrough cell C23. The autocorrelation coefficient of the first differences is calculatedin cell D4 as 0.184, which is smaller than the two-standard-error value of 0.447calculated in cell D7. This, combined with the apparent statistical stationarity wesee in the time series plot of the differences in Figure 11.7 lets us conclude that thedifferences can be modeled with Crystal Ball as though they are IID.

To generate potential future values of the sales time series, we used CrystalBall’s distribution-fitting procedure to fit a Triangular(-69.54,17.99,100.93) distri-bution to the values in cells C5:C23 and used that distribution to specify CrystalBall assumptions in cells C25:C29. The values in B25:B29 are calculated usingExpression 11.3. Cell B25 has the formula =B23+C25. Cell B26 has the formula=B25+C26, and this was copied and pasted to cells B27 and B28.

You can forecast as many steps ahead as desired using the random walkmodel, but realize that in doing so you are assuming implicitly that the distributiongenerating the differences remains stationary over the future period for which yougenerate values. The adequacy of this assumption depends on the context. It maywell be adequate for a few steps ahead, but the variance of the random walk modelincreases linearly with time, so for prolonged use of the model you will want to updatethe model by fitting distributions to the new data value changes as you observe them.


FIGURE 11.7 Crystal Ball model on the ‘‘Scalar Random Walk’’worksheet of RandomWalkWithDrift.xls for forecasting a timeseries with a random walk with drift process. Cells C25:C29 areCrystal Ball assumptions, and B25:B29 are Crystal Ballforecasts. Note that rows 8 through 22 are hidden.

Forecasting with Vector Random Walk Model

You can also use the random walk model to simulate observations from time seriesthat have both autocorrelation and correlation between series. This is illustrated inFigure 11.8 for the sales of three industrial products labeled X, Y, and Z in columnsB, C, and D. The procedure for forecasting more than one (that is, a vector) timeseries is similar to forecasting a single (scalar) time series. However, with a vectorrandom walk model, we take into account the correlation between changes in timeseries at the same time period as well as using the random walk model to induceautocorrelation among the levels of the time series.


FIGURE 11.8 Crystal Ball model on the ‘‘VectorRandom Walk’’ worksheet ofRandomWalkWithDrift.xls for forecasting a vector timeseries with a random walk with drift process.Cells E29:G33 are Crystal Ball assumptions, andB29:D33 are Crystal Ball forecasts. Note that rows 19through 27 are hidden.

In cells E5:G28 of Figure 11.8, we found the first differences of the X, Y, andZ time series in cells B4:B28. The autocorrelations in cells E35:G35 indicate thatthe differences follow a random process. Using Crystal Ball’s Batch Fit feature, wemodeled the changes in X, Y, and Z as Normal distributions with parameters thatyou will find in the file. Figure 11.9 shows the correlation matrix for the changes incells M11:O13.

Again, you can forecast as many steps ahead as desired using the vector randomwalk model, but realize that you are assuming implicitly that the random processesgenerating the differences remain stationary in regard to their distributions and theircross correlations.


FIGURE 11.9 Information generated byCrystal Ball’s Batch Fit tool on the firstdifferences of the X, Y, and Z times series infile RandomWalkWithDrift.xls.

MULTIPLICATIVE RANDOM WALK MODEL

If the time series of returns on a financial asset are IID, then we can use amultiplicative model to generate potential future prices of the asset. This is illustratedin Figure 11.10, which has data obtained from finance.yahoo.com. Cells B8:B170hold the monthly adjusted closing prices of the exchange traded fund (ETF) basedon the Standard & Poor’s 500 Composite Stock Price Index with sticker symbolSPY. Denote these prices as St for t = 1, . . . , 163. Note that the historical pricesare listed in reverse chronological order, with S163 in cell B8, down to S1 incell B170. In cells C8:C169, we have calculated the gross returns Rt = St/St−1 fort = 2, . . . , 163.

The multiplicative model used here to generate potential future prices is

St+1 = St × Rt+1, (11.4)

for t = 164, . . . , 168. These prices are calculated chronologically in cells G9:G13.The gross returns R164, . . . , R168 are calculated as Crystal Ball assumptions incells F9:F13. Each assumption is a lognormal distribution with mean 1.0088and standard deviation 0.0409, as determined by Crystal Ball’s distribution-fittingfeature. The lognormal distribution was chosen because of its adequate fit to the dataand its appealing property that it is bounded below by zero. This bound representswell the limited liability of owning shares of SPY, from which an owner cannot losemore than the total amount invested. The parameters of the fitted distribution tell usthat on average during the period February 1, 1993, through August 1, 2006, the ETFhad a monthly rate of return of 0.88 percent with a monthly standard deviation of4.09 percent. These monthly figures annualize to a mean rate of return of 12 × 0.88 =10.56 percent with a standard deviation of

√12 × 4.09 = 14.17 percent.

Cell G13 is defined as a Crystal Ball forecast, and its chart is shown inFigure 11.11. Using our methodology, a 95 percent certainty interval for the price


FIGURE 11.10 Crystal Ball model for forecastingwith a multiplicative model the January 2, 2007,adjusted closing value of SPY. Note that rows 15through 168 are hidden.

FIGURE 11.11 Crystal Ball forecast for the 2-Jan-07 adjusted closingvalue of SPY, based on the file SPY.xls. The actual closing value ofSPY on 3-Jan-07 was 141.37. Note that the new York StockExchange was closed on 2-Jan-07 in observance of a national day ofmourning for the death of former U. S. president Gerald R. Ford.


is from $111.02 to $158.17. Because we used the lognormal distribution for grossreturns, it is possible to have come up with this forecast analytically. However,we saw in Chapter 9 how the multiplicative model is used along with annualwithdrawals to come up with a retirement planning model for which a forecastcannot easily be obtained analytically. Ibbotson Associates (2006) describes wealthforecasting with Monte Carlo simulation and provides historical data from whichto estimate the necessary parameters.

The multiplicative model can also be used with assets whose returns arecorrelated within the same time period, but are serially uncorrelated. Figure 11.12is a model for three ETFs based on data for the period November 1, 2002, throughAugust 1, 2006, obtained from finance.yahoo.com. The adjusted closing price dataare in cells B8:D53 (not shown in Figure 11.12), and were used to calculate grossreturns in cells E8:G52. From these gross returns the lognormal distributions withmeans, standard deviations, and correlations indicated in cells I15:L22 were fitusing Crystal Ball’s Batch Fit tool. Again, see Chapter 9 for an example of how themultiplicative model was used for retirement planning.

FIGURE 11.12 Crystal Ball model for forecasting witha multiplicative model the January 2, 2007, adjustedclosing values of SPY, ADRA, and ADRU. The CrystalBall assumptions in cells J9:L13 have means andstandard deviations shown in cells I15:L17, and crosscorrelations shown in cells I19:L22. Cells M13:O13are Crystal Ball forecasts.


GEOMETRIC BROWNIAN MOTION MODEL

A special type of multiplicative random walk process is the geometric Brownianmotion (GBM) process, which is used widely for simulating stock prices. It is alsocalled exponentiated Brownian motion, and indeed to simulate GBM processeswe generate values from a Brownian motion process and then exponentiate them.Brownian motion takes its name from botanist Robert Brown, who observed in1827 that pollen particles floating in water under a microscope exhibited a ‘‘jittery’’motion even though they were inanimate. Work by Albert Einstein and others in theearly 1900s associated the normal distribution with the jittery movements observedby Brown.

In his 1900 doctoral thesis, Louis Bachelier described the movement of stockprices with what we now call Brownian motion in order to find the value of options.In the 1950s, economist Paul Samuelson rediscovered Bachelier’s thesis and went onto popularize the use of GBM as a model of stock prices and other financial assets.See Wilmott (2000) for a good, not-too-technical explanation of how GBM can bedeveloped from tossing coins.

Whereas it is possible for Brownian motion to take on negative values, GBM isalways positive because exponentiation always results in positive values. This is adesirable feature because the limited liability of stock ownership implies that pricescannot be negative. Also, it turns out that for GBM it is the percentage changesthat are IID rather than the absolute changes as in the additive random walk. Thisimplies that the stochastic percentage return is independent of the stock’s price levelfor GBM. This is an appealing feature. If an investor desires a 10 percent return onher investment, then with all else equal she will not care whether the 10 percent isearned from holding 3,000 shares of General Electric purchased for $30 per shareor one share of Berkshire Hathaway Inc. purchased for $90,000.

The non-negativity of the prices it generates, the independence of the percentagechanges, and the relative simplicity and good empirical fit all account for thepopularity of GBM for simulating stock prices. To learn more about the derivationof GBM, see Duffie (1996). To read more about its development and use in finance,see Rubinstein (2006).

Generating Stock Prices with GBM

To simulate stock prices using the GBM model, we generate independent replicationsof the stock price at time t + δ, from the formula

St+δ = Ste(µ−σ2/2)δ+σ√

δZ, (11.5)

where St is the stock price at time t, µ is the rate of return parameter stated on anannual basis, σ is the volatility parameter stated on an annual basis, δ is the timestep (in years), and Z is a standard normal random variate. The parameter σ is alsoknown simply as the volatility of the stock price, and is an important quantity in


FIGURE 11.13 Crystal Ball model for forecasting SPYprices with GBM.

modeling financial time series. Some authors refer to µ simply as the rate of return,but its interpretation takes some care as explained below.

Because ln(St+δ) is normally distributed, stock prices generated with GBM followa lognormal distribution. Figure 11.13 shows a model used to generate SPY priceswith GBM having µ = 9.49 percent and σ = 14.04 percent in cells G12 and G13,respectively. A forecast chart for the price of SPY on 2-JAN-07, is shown inFigure 11.14. We arrived at this price by generating prices in five steps of one month(δ = 1/12), but could have obtained similar results with just one step of five months(δ = 5/12) in Expression 11.5. The next section explains how the values of theparameters µ and σ were selected.

Estimating GBM Parameters

Given a time series of stock prices, S0, S1, S2, . . . , Sn observed at equally spaced timeperiods with interval δ (stated in years), we can estimate the values of µ and σ as


FIGURE 11.14 Forecast chart for SPY price on January 2, 2007. Alognormal density function is superimposed on the histogram.

FIGURE 11.15 Correlated GBM. Cells D4:E103 areassumptions. Cells B103 and C103 are forecasts.

was done in SPYwithGBM.xls. Define Ri as the gross return per period and ri as thecontinuously compounded rate of return per period on the stock. To estimate µ andσ , first find

Ri = Si/Si−1 and ri = ln(Ri) for i = 1, 2, . . . , n.


Using the standard formulas for sample mean and standard deviation, compute

r = 1n

n∑

i=1

ri and sr =√√√√ 1

n − 1

n∑

i=1

(ri − r).

Then the GBM parameters are estimated as

µ = rδ

+ s2r

2δand σ = sr√

δ

Note the difference between the GBM rate of return parameter µ and theexpected annual return on the stock. Suppose that stock price ST is generated by aGBM process with parameters µ and σ , starting at price St, where T > t and bothare stated in years. Then ST is lognormally distributed with mean and variance

E(ST) = Steµ(T−t)

Var(ST) = S2t e2µ(T−t)(eσ2(T−t) − 1)

For the SPY example, the expected mean and variance are 132.35 and 144.42,respectively. The sample mean and variance of the forecast in cell C178 are 134.54and 149.49, respectively. Furthermore, let r be the continuously compounded rateof return stated on an annual basis from time t to T, that is, ST = Ster(T−t). Then r isnormally distributed with mean and variance

E(r) = µ − σ 2

2

Var(r) = σ 2

(T − t)

This is verified by the simulation results in SPYwithGBM.xls.There are other methods for estimating volatility that use more information,

such as the daily high and low prices as well as the closing prices. See Wilmott(2000) and the references therein for more information.

Generating Correlated Stock Prices

Figure 11.15 shows part of a Crystal Ball model for generating correlated geometricBrownian motion in CorrelatedGBM.xls. This model lets you specify parametersfor means, standard deviations, and correlation for a market return and a stock. Italso calculates sample statistics for means, standard deviations, and correlation for amarket return and a stock, as well as showing time-series plots of stock and marketprices and a scatterplot of returns.


MEAN-REVERTING MODEL

The mean-reverting model is used for modelling commodity prices, foreign exchangerates, interest rates and other financial time series. Unlike the random walk withdrift model, which increases (or decreases) on average over time, the mean-revertingmodel has the characteristic that its level tends to fluctuate around the mean value,µ. One type of mean-reverting model is the autoregressive model, which takesadvantage of the autocorrelation in a time series to predict future values from pastvalues of the series.

AR(1) Process

An autoregressive (AR) model of order 1, known as an AR(1) model, is

Yt = φ0 + φ1Yt−1 + at, (11.6)

where at is a white noise series with mean zero and variance σ 2a . Expression 11.6 is

in the same form as a simple regression model, so we can use Excel’s Data Analysistool to find estimates of the parameters, φ0, φ1, and σ 2

a , then use these estimates tosimulate future values of the time series.

For example, the model in Figure 11.16 shows the market yield on 10-yearU.S. Treasury bonds for 2005. In order to use Excel’s regression capabilities, thedata in cells B4:B252 were copied and pasted to cells C5:C253. The values incolumn C are then called the Lag1 values simply because for each value of the seriesin rows 5 through 253 of column B, the previous value of the series is in the samerow of column C.

To estimate the parameters of the AR(1) model, mouse to Tools > Data Analysis. . . > Regression in Excel and you will get the dialog window shown in

Figure 11.17. By specifying B5:B253 as the Y Range and C5:C253 as the X Range,and checking the box next to Residuals, we generated a New Worksheet Ply named‘‘AR(1)’’ containing the output shown in Figures 11.18 and 11.19.

To simulate future values of the time series, we use the model

Yt = 0.132 + 0.969Yt−1 + et,

where et is Normal with mean zero and standard deviation 0.045. The values0.132, 0.969, and 0.045 are taken from cells B17, B18, and B7, respectivelyin Figure 11.18. To check whether the residuals appear to be white noise, theirautocorrelation is found in cell D25 to be 0.011, which gives a rule of thumb valueof 0.181 (much less than 2 in absolute value) in cell E25. Furthermore, using CrystalBall’s Distribution Fitting feature, we found that a normal(0,0.045) distribution fitthe residuals well.

For the AR(1) model specified in Expression 11.6, the mean is µ = φ0/(1 −φ1). Therefore, the mean to which our simulated data will revert is estimated as


FIGURE 11.16 Market yield on 10-year U.S. Treasury bonds for2005 obtained from www.federalreserve.gov/datadownload/ andtheir lag-1 values. Note that rows 7 through 249 are hidden.

0.132/(1 − 0.969) = 4.31. Although we can use this model to simulate future valuesfor an indefinite period, it may be best to reestimate the model parameters as newdata become available.

AR(p) Process

In the previous example, because the residuals in Figure 11.19 appeared to comefrom a white noise process, we concluded that an AR(1) process was suitable forsimulating future values. In this section, we consider what to do when the residualsfrom an AR(1) model do not appear to be from a white noise process.


FIGURE 11.17 Excel regression fitting an AR(1) model to the datadialog window used in Figure 11.16.

FIGURE 11.18 Excel regression output from fitting an AR(1) model to the datain Figure 11.16.


FIGURE 11.19 Residuals from fitting an AR(1) model to the data in Figure 11.16. Note that rows 30through 271 are hidden.

Mean-reverting processes can be simulated as AR(p) processes, where the orderp is identified from the data. The model for an AR(p) process is

Yt = φ0 + φ1Yt−1 + . . . + φpYt−p + at,

where at is a white noise series with mean zero and variance σ 2a as in the specification

of an AR(1) model. Observations from an AR(p) model will revert to the mean level,

µ = φ0/(1 − φ1 − . . . − φp).

One strategy for using an AR(p) model is to find the smallest order p suchthat the residuals from the AR(p) autoregression have no statistically significantfirst-order autocorrelation. This assumes that if the first-order autocorrelation is


FIGURE 11.20 Data and lagged values contained on the ‘‘Data’’ worksheet of thefile OneYearTreasuryYieldsData.xls. Note that rows 21 through 206 are hidden.

zero, then higher orders will be zero also. While not always a good assumption, thismethod is useful for generating potential future values from many time series usedin financial models. For information about performing more thorough time seriesanalyses of financial time series for inferential purposes, see Tsay (2002).

An example of identifying and using an AR(p) model is shown in the fileOneYearTreasuryYields.xls for 1-Year Treasury Constant Maturity Rate datashown in cells B16:B207 of the ‘‘Data’’ worksheet in the Excel file shown inFigure 11.20. The LAG1, LAG2, and LAG3 values were placed in Columns B, C,and D, respectively, on the ‘‘Data’’ worksheet. Worksheets ‘‘AR(1)’’, ‘‘AR(2)’’, and‘‘AR(3)’’ show the output from fitting AR models of order 1, 2, and 3 to the timeseries values. Based on this output, we would select an AR(2) model for simulatingpotential future values. Figure 11.21 shows some of the output on the ‘‘AR(2)’’worksheet.

We reach this conclusion because the residuals from the AR(1) model haveautocorrelation of 0.462, which is large enough by our rule of thumb (because


FIGURE 11.21 Worksheet ‘‘AR(2)’’ of the file OneYearTreasuryYieldsData.xls.

6.35 > 2) to conclude that they are not uncorrelated. However, the residuals fromthe AR(2) model have autocorrelation of 0.004 (see cell C22 in Figure 11.21), whichis small enough by the rule of thumb value (0.053 in cell D22) to conclude thatthey come from a white noise process. We reach the same conclusion by noting thatthe estimate of φ3 in the AR(3) model, 0.010, is not statistically different from zerobecause its P-value in the Excel output is 0.891, which is much greater than theusual comparison value of 0.05.

In cells B208:B213 of the ‘‘Data’’ worksheet, we see how to use the mean-reverting AR(2) model to simulate data for the first six months of 2006 with theequation

Yt = 0.075 + 1.442Yt−1 − 0.461Yt−2 + et

where et is Normal with mean zero and standard deviation 0.200.This chapter covered some of the basic models used in financial modeling and

risk analysis. For more advanced models used for specialized purposes, see Tsay(2002), Fan and Yao (2003), or McNeil, Frey, and Embrechts (2005).

CHAPTER 12Financial Options

A financial option is a security that grants its owner the right, but not the obligation,to trade another financial security at specified times in the future for an agreed

amount. The financial security that can be traded in the future is called the underlyingasset, or simply the underlying. An option is an example of a derivative security,so named because its value is derived from that of the underlying. The problem ofplacing a value on an option is made difficult by the asymmetric payoff that arisesfrom the option owner’s right to trade the underlying in the future if doing so isfavorable, but to avoid trading when doing so is unfavorable.

Options allow for hedging against one-sided risk. However, a prerequisite forefficient management of risk is that these derivative securities are priced correctlywhen they are traded. Nobel laureates Fischer Black, Robert Merton, and MyronScholes developed in the early 1970s a method to price specific types of optionsexactly, but their method does not produce exact prices for all types of options. Inpractice, Monte Carlo simulation is often used to price derivative securities. Thischapter shows how to use Crystal Ball for option pricing.

The optionality leads to a nonlinear payoff that is convolved with the log-normally distributed stock price to result in a probability distribution for optionvalue that is difficult for many analysts to visualize without Crystal Ball. Thepayoff diagrams familiar to options traders give the range and level of optionvalue as a function of stock price but don’t offer insights into the probabili-ties associated with payoffs. However, Crystal Ball forecasts do this readily. Thenext section provides brief background material on financial options. For moreinformation, consult the books by Hull (1997), McDonald (2006), or Wilmott(1998, 2000).

TYPES OF OPTIONS

Denote the price of the underlying asset by St, for 0 ≤ t ≤ T, where T is theexpiration date of the option. The agreed amount for which the underlying is tradedwhen the option expires is called the strike price, which is denoted by K. There aremany different types of options. Some basic types are listed below.

170

Financial Options 171

Call. A call option gives its owner the right to purchase the underlying for thestrike price on the expiration date. The payoff for a call option with strikeprice K when it is exercised on date T is max (ST − K, 0).

Put. A put option gives its owner the right to sell the underlying for the strikeprice on the expiration date. The payoff for a put option with strike price Kwhen it is exercised on date T is max (K − ST, 0).

European. A European option allows the owner to exercise it only at thetermination date, T. Thus, the owner cannot influence the future cash flowsfrom a European option with any decision made after purchase.

American. An American option allows the owner to exercise at any time onor before the termination date, T. Thus, the owner of an American call (orput) option can influence the future cash flows with a decision made afterpurchase by exercising the option when the price of the underlying is high(or low) enough to compel the owner to do so.

Exotic. The payoffs for exotic options depend on more than just the price ofthe underlying at exercise. Some examples of exotics are: Asian options,which pay the difference between the strike and the average price of theunderlying over a specified period; up-and-in barrier options, which paythe difference between the strike and spot prices at exercise only if theprice of the underlying has exceeded some prespecified barrier level; anddown-and-out barrier options, which pay the difference between the strikeand spot prices at exercise only if the price of the underlying has not gonebelow some prespecified barrier level.

New types of options appear frequently. Because they are designed to coverindividual circumstances, analytic methods to price new derivative securities are notalways available when the securities are developed. However, it is possible to obtaingood estimates of the value of most any type of option using Crystal Ball and theconcept of risk-neutral pricing.

RISK-NEUTRAL PRICING AND THE BLACK-SCHOLES MODEL

Arbitrage is the purchase of securities on one market for immediate resale on anotherin order to profit from a price discrepancy. Because the sale of the security in thehigher-price market finances the purchase of the security in the lower-price market,an arbitrage opportunity requires no investment capital. An arbitrage opportunity issaid to exist when a combination of trades is available that requires no investmentcapital, cannot lose money, and has a positive probability of making money for thearbitrageur.

In an efficient market, arbitrage opportunities cannot last for long. As arbi-trageurs buy securities in the market with the lower price, the forces of supply anddemand cause the price to rise in that market. Similarly, when the arbitrageurs sellthe securities in the market with the higher price, the forces of supply and demand


cause the price to fall in that market. The combination of the profit motive andnearly instantaneous trading ensures that prices in the two markets will convergequickly if arbitrage opportunities exist.

Using the assumption of no arbitrage, financial economists have shown that theprice of a derivative security can be found as the expected value of its discountedpayouts when the expected value is taken with respect to a transformation of theoriginal probability measure called the equivalent martingale measure or the risk-neutral measure. See Duffie (1996), Hull (1997), McDonald (2006), or Wilmott(1998, 2000) for more about risk-neutral pricing.

The price of a fairly valued European put option is the expected present valueof the payoff E

[e−rTmax (K − ST, 0)

], where the expectation is taken under the risk-

neutral measure. To compute this expectation, Black and Scholes (1973) modeled thestochastic process generating the price of a non-dividend-paying stock as geometricBrownian motion (GBM).

The Black-Scholes price for a European call option on a non-dividend-payingstock trading at time t is

Ct(St, T − t) = StN(d1) − Ke−r(T−t)N(d2), (12.1)

where

d1 = log(St/K) + (r + 1

2σ 2)

(T − t)

σ√

T − t, (12.2)

d2 = log(S/K) + (r − 1

2σ 2)

(T − t)

σ√

T − t= d1 − σ

√T − t, (12.3)

N(di) is the cumulative distribution value for a standard normal random variablewith value di, K is the strike price, r is the risk-free rate of interest, and T is the timeof expiration.

The Black-Scholes solution for a European put option on a non-dividend-payingstock trading at time t is:

Pt(St, T − t) = −StN(−d1) + Ke−r(T−t)N(−d2), (12.4)

where d1 and d2 are given by expressions (12.2) and (12.3) above.Note that the variables appearing in the Black-Scholes equations are the current

stock price, St; stock price volatility, σ ; strike price, K; time of expiration, T; andthe risk-free rate of interest, r; all of which are independent of individual riskpreferences. This allows for the assumption that all investors are risk neutral, whichleads to the Black-Scholes solutions above. However, these solutions are valid in allworlds, not just those where investors are risk neutral.


Option Pricing with Crystal Ball

In the Black-Scholes worldview, a fair value for an option is the present value ofthe option payoff at expiration under a risk-neutral random walk for the underlyingasset prices. Therefore, the general approach to using simulation to find the price ofthe option is straightforward:

1. Using the risk-free measure, simulate sample paths of the underlying statevariables (e.g., underlying asset prices and interest rates) over the relevant timehorizon.

2. Evaluate the discounted cash flows of a security on each sample path, asdetermined by the structure of the security in question.

3. Average the discounted cash flows over sample paths.

In effect, this method computes an estimate of a multidimensional integral thatyields the expected value of the discounted payouts over the space of sample paths.The increase in complexity of derivative securities has led to a need to evaluatehigh-dimensional integrals. Monte Carlo simulation is attractive relative to othernumerical techniques because it is flexible, easy to implement and modify, and theerror convergence rate is independent of the dimension of the problem.

To simulate stock prices using the assumptions behind the Black-Scholes model,generate independent replications of the stock price at time t + δ from the formula

S(i)t+δ = St exp

((r − σ 2/2)δ + σ

√δZ(i)

), (12.5)

for i = 1, . . . , n, where St is the stock price at time t, r is the riskless interest rate, σ

is the stock’s volatility, and Z(i) is a standard normal random variate.The Excel files EuroCall.xls in Figure 12.1 and EuroPut.xls in Figure 12.2 con-

tain simulation models for pricing European Call and Put options on a stock withcurrent price S0 =$100 and annual volatility σ = 40%. The options have strike priceK =$100, and six months until expiration, in a world with risk-free rate r = 5%.Of course, these are securities for which the Black-Scholes formulas (12.1) and(12.4) provide an exact answer, so there is no need to use simulation to price them.However, European options serve to help us see how well the Monte Carlo pricingapproach works—since we know the exact solution, it becomes possible to checkthe accuracy of our simulation results against the exact solution provided by Black-Scholes. In the Excel file EuroCall.xls, the European call price estimated by simulationwith 100,000 iterations is $12.33 (with standard error 0.06), while the Black-Scholesprice is $12.39. In EuroPut.xls, the European put price estimated by simulation with100,000 iterations is $9.92 (0.04), while the Black-Scholes price is also $9.92.

The increased availability of powerful computers and easy-to-use software hasenhanced the appeal of simulation to price derivatives. The main drawback ofMonte Carlo simulation is that a large number of replications may be required toobtain precise results. Fortunately, computer speeds have increased greatly in the last


FIGURE 12.1 Spreadsheet segment from model to simulate aEuropean call option.

FIGURE 12.2 Spreadsheet segment from model to simulate aEuropean put option.

30 years and software algorithms such as Crystal Ball’s Extreme Speed feature havebecome more efficient. Furthermore, variance reduction techniques can be appliedto sharpen the inferences and reduce the number of replications required. Variancereduction techniques are covered in Appendix C.

PORTFOLIO INSURANCE

In this section, we use Crystal Ball to simulate the combination of holding a putoption with the underlying asset. This limits the upside potential, but protectsagainst potential losses and so is a form of portfolio insurance. We will see how thisstrategy lowers the risk and expected value from the levels obtained when holding


FIGURE 12.3 Spreadsheet segment from model inVFH.xls to simulate the return on holding a stock and aput option.

the underlying asset by itself. Although this strategy lowers risk for any selectedunderlying asset, it might induce a money manager to purchase a riskier underlyingwith a higher expected return if it can be protected with a put.

Figure 12.3 shows a spreadsheet segment from the model in file VFH.xls usedfor estimating the return on a portfolio composed on August 21, 2006, of oneshare of the exchange-traded fund (ETF) tracking stock VFH and a put option onVFH that expires on March 16, 2007. The holding period is calculated as 0.57years in cell E13. VFH tracks the performance of the Morgan Stanley (MSCI) U.S.Investable Market Financials benchmark index. This index consists of stocks oflarge, medium-size, and small U.S. companies within the financial sector, which iscomposed of companies involved in activities such as banking, mortgage finance,consumer finance, specialized finance, investment banking and brokerage, assetmanagement and custody, corporate lending, insurance, financial investment, andreal estate. The drift and volatility parameters were estimated as 11.50 percent and11.75 percent, respectively, from the monthly closing prices of VFH for the previous31 months. Cell D21 has the rate of return earned if the stock alone was held from


FIGURE 12.4 Forecast charts from model in VFH.xls to simulate thereturn on holding a stock and a put option.

August 21 through March 16, and cell D23 has the rate of return earned on theportfolio of the stock and the put option held during the same period.

Figure 12.4 shows the forecast charts for cells D21 and D23, specified to havethe same scale on the horizontal axes. Note how the option to sell VFH for theexercise price, if its price falls below that, limits the downside value of the portfoliobut not the upside. However, this protection comes at the cost of the price of theoption, so the mean percentage return on the portfolio of 4.07 percent is lower than


that on holding the stock alone, which is 6.71 percent. This is similar to buyinginsurance coverage to protect against a loss, so the strategy of purchasing a putalong with stock is a form of portfolio insurance.

AMERICAN OPTION PRICING

Whereas a European option grants its holder the right, but not the obligation, tobuy or sell shares of a common stock for the exercise price, K, at expiration time T,an American option grants its holder the right, but not the obligation, to buy or sellshares of a common stock for the exercise price, K, at or before expiration time T.The Black-Scholes expressions (12.1) and (12.4) are for European options and thusyield approximations for the values of American call and put options. In practice,numerical techniques are used to obtain closer approximations of options that canbe exercised at or before expiration time T.

The fair value of an American put option is the discounted expected value ofits future cash flows. The cash flows arise because the put can be exercised at thenext instant, dt, or the following instant, 2dt, if not previously exercised, . . . , adinfinitum. In practice, American options are approximated by securities that canbe exercised at only a finite number of opportunities, k, before expiration at timeT. These types of financial instrument are called Bermudan options. By choosing klarge enough, the computed value of a Bermudan option will be practically equal tothe value of an American option.

Geske and Johnson (1984) develop a numerical approximation for the valueof an American option based on extrapolating values for Bermudan options havingsmall numbers (1, 2, and 3) of exercise opportunities. Their results are exact inthe limit as the number of exercise opportunities goes to infinity. Broadie andGlasserman (1997) used simulation to price American options by generating twoestimators, one biased high and one biased low, both asymptotically unbiased andconverging to the true price. Avramidis and Hyden (1999) discuss ways to improvethe Broadie and Glasserman estimates. Longstaff and Schwartz (1998) provide analternate method for pricing American options.

The early exercise feature of American options makes their valuation moredifficult because the optimal exercise policy must be estimated as part of thevaluation. This free–boundary aspect of the pricing problem led some authors toconclude that Monte Carlo simulation is not suitable for valuing American options(for example, Hull 1997). However, we’ll see next how to use Crystal Ball andOptQuest for this purpose.

The file BermuPut.xls contains an example of valuing an Bermudan put optionwith initial stock price S0 = 40, risk-free rate r = 0.0488, time to expiration T =0.5833 (seven months), volatility σ = 0.4, strike price K = 45, and six early-exerciseopportunities at Months 1 through 6. From Geske and Johnson (1984), the truevalue of this option is $7.39.

The spreadsheet in Figure 12.5 illustrates a method to price this option usingCrystal Ball and OptQuest. This method uses OptQuest’s tabu search to identify an


FIGURE 12.5 Spreadsheet segment from model to simulatea Bermudan put option.

FIGURE 12.6 Forecast from model to simulate a Bermudan putoption. The values of the decision variables in cells E12:E17 wereselected by OptQuest.


FIGURE 12.7 Constraints from model to simulate a Bermudanput option.

optimal policy, then a final set of iterations to estimate the value of the option underthe identified policy. The estimated price for the option described above is shown inFigure 12.6 as $7.42. The standard error of this estimate is $0.06.

Figure 12.7 shows the only constraints on the decision variables. Because thelonger the time left until expiration, the greater the chance of the stock price fallingbelow the exercise price, so the early-exercise boundary value should also be lessthan the value at a later time. These constraints are imposed in Figure 12.7 byrequiring the bound at month t to be greater than or equal to the bound in theprevious month, t − 1, for t = 2, 3, 4, 5, 6.

EXOTIC OPTION PRICING

Exotic options are financial instruments having more complicated payoff structuresthan ‘‘plain vanilla’’ puts and calls. As the term exotic is used to describe options inthe sense of unusual, there is not a well-accepted categorization of exotic options.What are exotic options to one trader may be traded on a daily basis by another,and therefore not unusual. For our purposes, we use the term to apply to any optionother than the European or American puts and calls we have described thus far.

There are far too many exotic options to list here, but the next three subsec-tions show how to model some options that are representative of those you mightencounter.

Digital optionsDigital options pay either a prespecified amount of an asset, or nothing at all. Forexample, a European cash-or-nothing digital (also called a binary) call option pays$1 if and only if the price of the underlying exceeds the strike price on the exercisedate. That is, the payoff is

$1 if ST > K,0 otherwise.


FIGURE 12.8 Spreadsheet segment from model inAssetOrNothingCall.xls to simulate the return on anasset-or-nothing call option.

A European asset-or-nothing digital call option pays one share of the underlyingasset if and only if the price of the underlying exceeds the strike price on the exercisedate. That is, the payoff is

ST if ST > K,0 otherwise.

Figure 12.8 shows a model to value an asset-or-nothing call option with strike price$105 expiring in one year. It can also be used to model a cash-or-nothing optionhaving payoff $1 by inserting into cell D11 the formula =IF(B12>K,1,0).

Barrier Options

On September 28, 1998, the New York Times reported that Sprint chairmanWilliam T. Esrey stood to earn call options having a strike price of $47.94 for one


FIGURE 12.9 Spreadsheet segment from model inEsreyOptions.xls to simulate the return on an up-and-in barriercall option.

million shares of Sprint stock if the stock price reached a barrier price of $95.875sometime in the future. On September 30, 1998, Sprint’s stock price closed at$72.00. The file EsreyOptions.xls in Figure 12.9 contains a model to estimate thevalue of Mr. Esrey’s up-and-in barrier call options on December 31, 2000, basedon the historical drift and volatility of Sprint stock estimated from monthly closingprices during the period January 31, 1996, through September 30, 1998, which wasa period of very dynamic growth in the telecommunications industry.

Asian Options

Figure 12.10 shows a model used to determine the price of an Asian average-pricecall option for a stock with S0 = $40, K = $40, σ = 30%, r = 8%, and T = 0.25.


FIGURE 12.10 Spreadsheet segment from model in AsianCall.xls tosimulate the return on an average-price call option.

The value of $1.98 (with a standard error of $0.03) is consistent with McDonald(2006), who gets a price of $2.03 ($0.03).

Denote the price of the stock at time t as St. Then for this option, which paysoff on the arithmetic average of the monthly prices, the option price is found bysimulating the stock prices S1, S2, and S3, then taking the mean over all iterations ofthe quantity

e−rTE (max [(S1 + S2 + S3)/3 − K, 0]) .

Analytic solutions exist for pricing Asian options that pay off on the geometricaverage (see McDonald 2006). To price this as a geometric Asian option with Crystal


Ball, simulate stock prices S1, S2, and S3, then take the mean over all iterations ofthe quantity

e−rTE(max

[3√

S1S2S3) − K, 0])

.

BULL SPREAD

Options traders often hold more than one option on the same stock. This is calledan option strategy. Figure 12.11 shows a model for a bull-spread strategy in whicha trader buys a call with strike price K = $130 and writes a call with strike priceK = $140. Both calls expire on December 15, 2006.

Figure 12.11 shows a mean return of −5.96 percent on the bull spread if theassumed µ and σ parameters of the stock are 5 percent and 11 percent, respectively.

FIGURE 12.11 Spreadsheet segment from model inBullSpread.xls to simulate the return on a bull spread.


FIGURE 12.12 Decision table from model inBullSpread.xls to simulate the return on a bull spreadfor several values of µ and σ .

Because different traders have different expectations for µ and σ , Figure 12.12shows estimates of the bull-spread strategy’s mean return as a function of differentlevels of µ and σ . In general, the mean return increases as a function of bothparameters.

PRINCIPAL-PROTECTED INSTRUMENT

While not strictly an option, the analysis of principal-protected instruments (PPIs)is included here to demonstrate how to model another derivative security recentlyintroduced in the marketplace.

PPIs are sold to risk-averse investors who wish to contractually guarantee thatthey will not lose any of their initial investment, but also wish to participate tosome extent in upward movements of the price of a financial investment. They arehybrid securities that combine a fixed income instrument with a series of derivativecomponents. PPIs have been engineered for assets such as equities, currencies, interestrates or commodities.

Figure 12.13 shows a model for valuing a PPI with the following characteristics:For every quarter of its five-year life, the PPI quarterly return tracks the quarterlyreturn on the underlying asset XYZ. However, if the quarterly rate of return onXYZ exceeds 15 percent for any quarter, the PPI return for that quarter is cappedat 15%. At the end of five years, the PPI will deliver a final amount determined bythe 20 quarterly returns specified in the contract.


FIGURE 12.13 Model to compute distribution of rates ofreturn on a principal-protected instrument.

Denote the initial investment as I, the final value of XYZ as Fxyz, the final valueof PPI as Fppi, and the quarterly rate of return on XYZ stock as Ri for i = 1, 2, . . . , 20.The final value of XYZ is

Fxyz = I20∏

i=1

(1 + Ri),

while the final value of PPI is

Fppi = max

[I

20∏

i=1

min(1 + Ri, 1.15), I

].


FIGURE 12.14 Distribution of the difference in annualized rates ofreturn on a PPI and the underlying asset.

The simulation model in Figure 12.13 generates quarterly values for asset XYZ usinggeometric Brownian motion with parameters µ = 12 percent and σ = 30 percent.The ‘‘Report’’ worksheet shows the final values and annualized rates of return onholding the PPI and on XYZ. Figure 12.14 shows the difference in annualized ratesof return when holding the PPI and XYZ alone. The risk-averse investor pays 2.87percent on average to guarantee that the principal is not lost. Figure 12.14 alsoshows that the probability is about 70 percent that the investor would realize agreater return by holding XYZ alone than by holding the PPI.

CHAPTER 13Real Options

T his chapter describes a recent topic in finance called real options analysis (ROA)and shows how Crystal Ball and OptQuest can help you determine the value of

real options. As we have seen, a financial option is the right, but not the obligation,to buy (or sell) an asset at some point within a predetermined period of time fora predetermined price. ROA is used as an alternate methodology for evaluatingcapital investment decisions involving a high degree of managerial flexibility, suchas research and development projects or new product decisions. Unlike the simplenet present value (NPV) method used in traditional finance theory, ROA treats aninvestment opportunity as either a single option or a compound option (a sequence ofoptions). The traditional NPV method does not value managerial flexibility correctlywhen it relies on the false assumption that the investment is either irreversible orthat it cannot be delayed.

In this chapter, we will see the similarity between financial and real options,then discuss applications of ROA and some analytical methods that have beenused with real options. The real option valuation (ROV) tool described in the finalsections combines the use of Crystal Ball and OptQuest to determine the value ofopportunities that contain real options.

FINANCIAL OPTIONS AND REAL OPTIONS

With a financial option the initial investment in an option contract buys thepotential opportunity to enjoy positive cash flow when future spot price changes ofthe underlying financial asset favor doing so, but does not carry the obligation torealize negative cash flow if unfavorable conditions prevail. For example, the holderof a call option is not obligated to purchase the underlying at the strike price if itsspot price is below the strike price on the expiration date, and the holder of a putoption is not obligated to sell the underlying at the strike price if the spot price isabove the strike price on the expiration date. This flexibility to limit one’s lossesadds value to a financial option contract when there is uncertainty about the futurespot price of the underlying.

Contrast the flexibility of an option contract to a futures contract, which specifiesa price and a future date for a transaction that both parties are obligated to complete.

187


For example, if you are to be paid a fixed amount of Indian rupees (INR) one yearfrom now, but you want to lock in the amount of American dollars (USD) you willgain at that time, you can enter into a futures contract (at some cost to you) thatspecifies an exchange rate for the amount of USD to receive in exchange for INR oneyear from now. Once you are locked into the exchange rate, you are shielded fromfluctuations in the USD/INR spot exchange rate. If the spot exchange rate is lowernext year than the rate you locked in, you will end up with more USD than youwould otherwise receive at the spot exchange rate, but if the spot exchange rate ishigher next year, you will end up with fewer USD than you would otherwise. Witha futures contract, you bear the risk of losing more than just the cost of the contractif the USD/INR exchange rate rises—you also lose the opportunity to benefit fromthe higher exchange rate.

With a rupee put option contract, you can simply choose not to complete thetransaction if the spot exchange rate exceeds the strike price. You will lose the costof entering into the option contract, but you will benefit from selling your INR atthe higher spot exchange rate. With all else equal, an option contract is worth morethan a futures contract because an option contract offers more flexibility than afutures contract. Chapter 12 describes how to use Crystal Ball to determine optionvalues. For more information about options and futures contracts, see McDonald(2006) or Wilmott (2000).

With a real option—an option on a real asset—the initial investment relatedto the asset buys the potential opportunity to continue, expand, or abandon theuse of the asset when it is favorable to do so, but does not carry the obligationto realize some losses when unfavorable conditions prevail. Because efforts suchas testing potential oil-drilling sites can be viewed as learning options, financialmodels similar to those used for determining financial option values can be used todetermine the value of the real options embedded in the opportunity to test for oilat a particular site.

To learn more about the theory underlying real options, see the texts by Dixitand Pindyck (1994), or Trigeorgis (1996), which summarize much of the early workdone in applying financial options valuation methodology to real options problems.The next section describes how real options have been applied in various contexts.

APPLICATIONS OF ROA

For a good, nontechnical introduction to real options analysis, see Copeland andKeenan (1998a, 1998b), who categorize real options into the three broad categoriesdescribed below.

1. Investment/growth options. These include (1) scale-up options, where earlyentrants can scale up later through sequential investments as their market grows;(2) switch-up options, where speedy commitments to the first generation of aproduct or technology give managers a preferential position to switch to thenext generation of the product or technology; and (3) scope-up options, where

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investments in proprietary assets in one industry enables managers to enteranother industry with a competitive cost advantage.For example, a venture capitalist (VC) who invests in stages uses ROA ofthe growth option to value a start-up company. By structuring the contractproperly, the VC retains exclusive rights to a portion of the profits from thestart-up venture. However, if the VC decides later not to invest further, any lossis limited to the amount already invested. The VC is not obligated to pay thestart-up’s debts if the venture fails.

2. Deferral/learning options. Also called study/start options, these are oppor-tunities to delay investment until more information or skill is acquired. Forexample, an oil company uses ROA to evaluate exploration investment strate-gies, in which drilling sites undergo various types of testing before the decisionwhether or not to drill is made. A pharmaceutical firm uses ROA to evaluatedrug development projects, in which investments are made in several phases ofexperimentation with the drug compound before seeking regulatory approvaland going to market.

3. Disinvestment/shrinkage options. These include (1) scale-down options, wherenew information that changes the expected payoffs can cause managers to shrinkor shut down a project before completion; (2) switch-down options, wheremanagers have the ability to switch to more cost-effective and flexible assets asnew information is obtained; and (3) scope-down options, where the scope ofoperations is decreased or even ceased when managers see no further potentialin a business opportunity.For example, a manufacturing firm uses ROA to evaluate three types of powergenerators that use (1) natural gas, (2) fuel oil, or (3) both for fuel. The highercost of a dual-fuel generator may be offset by future savings obtained when thecost per energy unit of natural gas is lower than fuel oil, or vice versa. ROA candetermine a value for the flexibility to use the cheaper fuel when the dual-fuelgenerator is installed.

Myers (1984) is often credited with being the first to publish in the academicliterature the notion that Black and Scholes (1973) results could be applied to strate-gic issues concerning real assets rather than just financial assets. In the practitionerliterature, Kester (1984) suggested that the discounted cash flow valuation methodsin use at that time ignored the value of important flexibilities inherent in manyinvestment projects and that methods of valuing this flexibility were needed. ROA ismost effective when competing projects have similar values obtained with the simpleNPV method.

One difficulty in applying ROA is that real asset investments are usually affectedby more than one source of uncertainty, whereas all of the uncertainty drivingfinancial options is characterized by the volatility in spot prices of the underlyingfinancial asset. As we saw in Chapter 12, the historical volatility of a financial assetis readily obtained from publicly available market prices. Options with values driven


by multiple sources of uncertainty are called rainbow options. Combinations ofrainbow and learning options often exist in practice.

Thinking about investment projects in option terms encourages managers todecompose an investment into its component options and risks, which can leadto valuable insights about sources of uncertainty and how uncertainty will resolveover time (Brabazon 1999). Options thinking also encourages managers to considerhow to enhance the value of their investments by building in more flexibility wherepossible. Bowman and Moskowitz (2001) suggest that ROA is useful because itchallenges the type of investment proposals that are submitted and encouragesmanagers to think proactively and creatively.

ROA has the potential to allow companies to examine programs of capitalexpenditures as multi-year investments, rather than as individual projects (Copeland2001). Such programs of investments are strategic and highly dependent on marketoutcomes, which is just the decision climate under which Miller and Park (2002) findROA to be most useful. However, ROA and NPV are complementary techniques,with NPV being suitable for basic replacement decisions.

Early work on real options valuation suggests that if the analogous real optionsparameters can be estimated, any method used to value financial options canpotentially be used to value real options. Often though, many of the assumptionsmust be relaxed to make the connection. Amram and Kulatilaka (1999), Copelandand Antikarov (2001), and Mun (2002) provide guidelines for analyzing real optionswith financial-option pricing techniques. The remainder of this section describes twoearly techniques for ROA: the Black-Scholes method, and lattice methods.

Black-Scholes Method

The Black-Scholes method relies on the assumption that project values follow ageometric Brownian motion (GBM) stochastic process. While useful in the abstract,GBM is difficult to use in practical real options problems involving many sourcesof uncertainty and interrelated decisions. In order to use this method, one mustsomehow encapsulate the random effects of all the important real-world compli-cations into one summary measure—the volatility parameter of the GBM process.Relatively few managers have the background or inclination to estimate the values ofthe volatility parameters that are necessary for using Black-Scholes formulas to valuecomplicated real options in industry. However, the Black-Scholes model is usefulfor gaining insights into real options valuation and how projects can be managed toincrease their real option value.

Lattice Methods

Lattice methods also rely on the assumption that project values follow a GBMstochastic process. While the equations used in lattice methods are perhaps easierto grasp than those underlying Black-Scholes, lattice methods are simply a wayto approximate a GBM process and thus suffer from the same limitations as

Real Options 191

Black-Scholes—namely, that so many important real-world complications must beencapsulated in the volatility parameter. Hence, many managers are uncomfortablewith the estimation of the volatility parameters necessary to use lattice methods forROA in industry. However, those trained in finance theory may well be comfortableusing this technique. Mun (2002) has developed software for evaluating real optionswith lattice models that Decisioneering markets as the Real Options Analysis Toolkit.

BLACK-SCHOLES REAL OPTIONS INSIGHTS

The Black-Scholes model provides insights into the factors affecting the value of realoptions and how managers can manage their opportunities to increase this value. Tosee this, consider the Black-Scholes formula for a European call option on a stockthat pays dividends at the continuous rate δ:

C(S, K, σ , T, δ, r) = Se−δTN(d1) − Ke−rTN(d2), (13.1)

where

d1 = ln(S/K) + (r − δ + 12σ 2)T

σ√

T(13.2)

d2 = d1 − σ√

T (13.3)

and N(x) is the cumulative normal distribution function, which is the probabilitythat a number drawn randomly from the standard normal distribution (i.e., a normaldistribution with mean 0 and variance 1) will be less than x.

The Black-Scholes formula for a European put option on a dividend-payingstock is

P(S, K, σ , T, δ, r) = Ke−rTN(−d2) − Se−δTN(−d1), (13.4)

where N(x) is the cumulative normal distribution function, and d1 and d2 are givenby equations (13.2) and (13.3).

According to the Black-Scholes option-pricing models (13.1) and (13.4), optionsderive their value from six main factors. These factors are most easily expressed interms of financial options, but the analogy to real options provides insights into thefactors associated with strategic investment decisions. The factors are:

Stock price, S. The value of the underlying stock on which an option ispurchased. This is the stock market’s estimate of the present value of allfuture cash flows arising from ownership of the stock. Its analog in areal options analysis is the present value of cash flows expected from theinvestment opportunity under consideration. Some examples of the sourcesof uncertainty that affect the present value of cash flows from investment


are: market demand for products and services, labor supply and cost, ormaterials supply and cost.

Exercise price, K. The predetermined price at which the option can be exer-cised. Its real options analog is the present value of all the investmentcosts that are expected over the lifetime of the investment opportunity. Theavailability, timing, and price of real assets to be purchased all affect theuncertainty in this parameter.

Volatility, σ . A measure of the unpredictability of stock price movements,usually expressed as the standard deviation of the growth rate of the valueof future cash flows associated with the stock. Its real options analog isa measure of uncertainty of the cash flows associated with the investmentopportunity. This uncertainty arises from volatility in market demand, laborsupply and cost, and materials supply and cost. The correlations betweenthese factors also affects the volatility parameter.

Time to expiration, T. The period during which the option can be exercised. Itsreal options analog is the period during which the investment opportunityis available. This period depends on the product life cycle, the firm’scompetitive advantages, and the contractual arrangements made by thefirm.

Dividends. Sums paid regularly to stockholders at a constant continuous rate,δ. Dividends reduce a financial option payoff when the option is exercisedafter a dividend payout, which reduces the stock value. Their real optionsanalogs are the expenses that drain away potential project value over theduration of the option. The cost of waiting could be high if competitorsenter the market. Thus, the cost of waiting to invest might be reducedby locking-in key customers, or lobbying for regulatory constraints whenpossible to discourage competitors from exercising their options to enter themarket.

Interest rate, r. The yield on financial securities with the same maturity asthe duration of the option. The risk-free rate of interest is used in theBlack-Scholes model, but a different rate might be appropriate for analternate option valuation method.

According to the Black-Scholes model, increases in stock price, volatility, timeto expiration, and interest rates increase financial option values, while increasesin exercise prices and dividends reduce financial option values. These qualitativerelationships are generally true for real options as well. See Leslie and Michaels(1997), who describe how to apply options thinking to strategic situations by usingthe qualitative relationships as guidelines for managerial action.

However, real options have additional features that distinguish them fromthe type of financial options for which the Black-Scholes model was derived. TheBlack-Scholes model is an exact solution to a pricing problem that was simplifiedto make it solvable. The main simplification is called the European feature of theoption, which means that the option is assumed to be exercisable at only a single

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time point in the future. Most financial and real options are said to have Americanfeatures, which means that those options can be exercised at any point in timebetween their purchase and expiration. The valuation of American-style options ismore difficult than the valuation of European options.

In practice, the difficulty introduced by the American exercise feature can beovercome partially by assuming a Bermudan feature, which means that an optioncan exercise at one of several discrete points between purchase and expiration(rather than continuously as with an American option). The Bermudan assumptionis consistent with ROA if the decisions to make investments will be implemented onlyat discrete times (e.g., quarterly). The real options valuation (ROV) tool describedin the next section uses Crystal Ball and OptQuest to value real options in a mannersimilar to the valuation of financial Bermudan options in Chapter 12. The ROV toolanalyzes real-options investment opportunities by modeling cash flows occurringover a period of time, punctuated by key decisions to be made by management aboutwhether to make additional investments, continue with no further investment, orabandon the investment opportunity.

ROV TOOL

The ROV tool is simply the use of Crystal Ball to add stochastic assumptions,decision variables, and forecasts to a deterministic spreadsheet, then finding theoptimal values of the decision variables using OptQuest. Thus, describing how touse the ROV tool serves as a summary of financial modeling and risk analysis withCrystal Ball. See Charnes, et al. (2004) for a description of how the ROV tool wasapplied in the telecommunications industry.

The tool is used by following the eight steps in Figure 13.1, which diagrams theROV modeling process. This process expands on the simulation modeling processdetailed in Chapter 3. Each step is explained next.

ROV Modeling Process

Step 1: Identify Options The first task in any ROV modeling effort is to identify theoptions in the problem in such a way that they can be modeled with decision variablesin a spreadsheet. If this cannot be done, Crystal Ball cannot be used to help you makea decision. However, because of the versatility and flexibility of spreadsheets, manyoption problems can be modeled with Crystal Ball. Next, be sure you can quantifythe uncertainty in the model’s variables and any statistical relationships betweenthem. Again, if this cannot be done, then building a spreadsheet ROV model is notpossible. While these two tasks might seem obvious, making sure at the outset thata Crystal Ball model can be used to help solve the problem is critical to the successof any ROV project.

Step 2: Build or Revise Model Be sure to design your model so that it will help solvethe problem you’ve identified. Again, this sounds obvious, but some analysts get so


Step 1Identify Options

Step 8Make Decision

Step 7Run OptQuest

Step 2 Build or Revise

Model

Step 3 Add or ReviseAssumptions

Step 4Run

Crystal Ball

Step 6SensitivityAnalysis

Step 5Analyze

Forecasts

FIGURE 13.1 ROV modeling process diagram.

caught up in the details of modeling that they lose sight of the big picture. Do notlet this happen to you.

Wherever possible, model the uncertain variables in the smallest componentfor which you have historical data collected. For example, suppose monthly salesrevenue is a variable in your model. If you have data collected on both unitssold and monthly sales revenue, in general it will be better to make units soldinto a Crystal Ball assumption rather than monthly sales revenue. Revenue can becalculated in the spreadsheet as units sold times price, and by breaking revenue intoits components, you have more flexibility by modeling the uncertainty in units soldrather than monthly sales revenue if you decide later to investigate a change in price,for example.

Another important point to keep in mind is to have each assumption includedonly once in your model, and have any calculations that depend on the assumption’svalue make reference to that cell. Novices sometimes put the same probabilitydistribution in two or more cells in a model, thinking that as long as the samedistribution—say a uniform(4000,6000), for example—is used in two places it willgive the same value in both places during a simulation trial. However, including adistribution in two places means that Crystal Ball will generate independent values ineach cell—for example, two different numbers drawn from the uniform(4000,6000)distribution—and the model will not represent the real-life situation the novice istrying to model.

You may also reach Step 2 in the process as the result of previous analyses.In particular, sensitivity analysis (Step 6) sometimes leads to changes in the model.This is both a natural and good thing to happen, because it usually means that theinsights you have gained are helping you to improve the model you are building.

Real Options 195

Some analysts build an initial model to work with for a while as a prototype,then throw it out and begin anew once they have a better understanding of thesituation. Sometimes it is better to start over with a redesigned model than tocontinue working with an inefficient design that you can’t bear to give up becauseyou’ve been working on it for so long. An alternate approach advocated by someauthors is to map out your spreadsheet on paper before you even open Excel. SeePowell and Baker (2007) for their take on this approach.

Step 3: Add or Revise Assumptions For novices, choosing a distribution and itsparameter values is usually the hardest part of simulation modeling. However,choosing which variables to make into assumptions and which to leave as deter-ministic can also be a challenge. Choosing the assumption variables is a matter ofusing your best judgment, intuition, and any data that you have available to identifythose you think are most important. After you have run the simulation you can usesensitivity analysis to measure the effect of each assumption on the forecast(s), andchange your initial choices later in the modeling process when appropriate.

The Crystal Ball tornado chart is used to measure the effect of changes in anyvariable (including deterministic variables) on a selected forecast. If you are havinga difficult time deciding which input variables should be probabilistic, and whichshould be deterministic, try using the tornado chart, which helps to identify the mostimportant variables in terms of impact on the forecasts.

If you have no idea of which distribution family to select from the distributiongallery, consider using the triangular or uniform distributions. By default, theparameters of these distributions will be set so that the mean of your assumption isequal to the simple value in the cell when you click the Define Assumption icon. Theminimum and maximum values will be set by default to 10 percent below the mean,and 10 percent above the mean, respectively. If no historical data are available, youcan ask a subject matter expert (e.g., an engineer, cost analyst, or project manager)to help you choose the parameters of a triangular or uniform distribution. See thedescriptions of these distributions in Appendix A for more information about settingthe parameters.

If you are fortunate enough to have historical data available for a variable usedin your model, you can have Crystal Ball analyze the historical data to suggest adistribution as described in Chapter 4. For some models, the nature of the processor underlying physics of the situation will suggest a distribution. See Appendix Afor specific examples of when each distribution might be used.

Step 4: Run Crystal Ball Click on Run > Single Step in the top menu of Crystal Ballto run just one iteration of the simulation. Look at the values of the assumptionsand forecasts to make sure they are realistic for your model. If they are not realisticvalues (meaning that they represent a combination of values that could not occur inreal life), then you have an error somewhere in your spreadsheet model.

Verify that your assumptions have the correct parameters, and that the Excelformulas are correct. Make any necessary changes, then use Single Step again to


check your changes. Repeat this process until you are comfortable with the results youget on each step. Once you have verified that your model is correct, make sure CrystalBall’s sensitivity analysis feature is turned on (click on Run > Run Preferences, thenclick the Options button, put a check in the box next to Calculate Sensitivity, andclick OK). Run the simulation for an initial number of trials. Try using 10,000 trials ifyou are using Extreme Speed (ES) mode. If you are unable to use ES mode because youhave a large, complicated model, try using at least 2,000 trials in Normal Speed mode.

Step 5: Analyze Forecasts Check the forecasts to see if they contain outcome valuesthat could occur in real life. Because the combined effects of the probabilisticassumptions can be very large, don’t be surprised if the range of outcomes is verywide. Click on Analyze > Extract Data. . . to extract the values generated by CrystalBall for the assumptions and the corresponding forecast values. Investigating theextreme points in a forecast and the assumption values that led to them can yielduseful insights.

Step 6: Sensitivity Analysis Click on Run > Open Sensitivity Chart in thetop menu to bring up the Sensitivity Chart. The model’s assumptions are listed onthis chart from top to bottom in descending order of the magnitude of their effectson the selected forecast. The magnitude of the effects is measured by the Spearmanrank correlation statistic (see Chapter 4). Use the sensitivity analysis informationto revise the assumptions (Step 3) or the model itself (Step 2). Begin with the topassumption listed on the chart, and work your way down. For each assumption,make sure you are satisfied that the distribution and its parameters represent thesituation adequately. Draw upon subject matter experts for guidance.

Step 7: Run OptQuest You might have to go through Steps 2–6 many times beforeyou are satisfied with the model. However, this will help you understand the problemmuch better. Many analysts claim that at this point of the process they feel like theyknow enough about the problem to make a decision just because they have studied itso intensely to get this far. However, when you are comfortable with the results, andhave obtained buy-in from the others involved in the decision-making process, youare ready to run OptQuest. Refer to Chapter 5 for the details of running OptQuest.

Step 8: Make Decision If the model has helped to completely solve the problem youfaced, congratulations! However, oftentimes the process of modeling leads to theidentification of other problems to solve. If so, begin the process again to solve thenew problem by returning to Step 1.

Value Added by Using ROV Tool

A major advantage of using Crystal Ball and OptQuest as the ROV tool is thatit can be applied to a large number of existing spreadsheet models. These existingmodels serve as ‘‘calculation engines’’ that are used by Crystal Ball to transform the

Real Options 197

stochastic inputs into random outputs for specified values of the decision variables.An analyst comfortable with the ROV tool can use it with existing spreadsheetmodels without necessarily having to understand all of the minute details of thecalculation engine. This makes the tool highly reusable, as it only requires theanalyst to be able to link the top-level worksheet to the calculation engine in existingspreadsheets.

In Figure 13.2, the calculation engine is represented by the existing spreadsheetsdepicted on the right side. The calculations that go into the determination of NPVare usually very complex and can involve links to many of the worksheets composingthe Excel model. Some high-level knowledge of the business case represented by thecalculation engine is required to make the link to the top-level worksheet. However,the ROV tool can be used with spreadsheets built by others if you understand howthe decision variables and stochastic assumptions are involved in the calculationof NPV. The function g(d1, d2, . . . , dk; a1, a2, . . . , an) in Figure 13.2 represents theresult of all the calculations taking place in the business case that lead to a valueof NPV for the decision variable values d1, d2, . . . , dk and the assumption valuesa1, a2, . . . , an. If you understand the calculation of the function g(·) well enough toknow how d1, d2, . . . , dk and a1, a2, . . . , an affect the calculation of NPV, then youcan use the ROV tool independently of the analysis leading to the construction ofthe calculation engine. This feature allows the tool to be used with any existing orfuture financial worksheets.

Decision Variablesd1, d2, dk

ROV Tool

Stochastic Assumptionsa1, a2, ..., an

Random Outputse.g., NPV

OptQuest finds the set of decisionvariable values {d1, d2, ..., dk}

that maximizeE[g(d1, d2, ..., dk; a1, a2, ..., an)]

NPV = g(d1, d2, ..., dk ; a1, a2, ..., an)

Existing Spreadsheets

CompanyBusiness Case

NPV Calculations

FIGURE 13.2 Depiction of links between the ROV tool and existing NPV calculations in yourcompany’s business cases. The function g(d1, d2, . . . , dk; a1, a2, . . . , an) represents the result of all thecalculations taking place in the business case that lead to a value of NPV for the decision variable valuesd1, d2, . . . , dk and the assumption values a1, a2, . . . , an.


Because the ROV tool is independent of the calculation engine, it is scalableto virtually any size desired. The only limits on the size of the model are thoseimposed by Microsoft Excel. Crystal Ball and OptQuest can handle a number ofdecision variables that is unlimited for most practical purposes. Note that the currentversion (available in 2006) of Extreme Speed mode takes longer to initialize whenthe business case is composed of many spreadsheets. For some complicated models,this initialization can take so long that you may be better off running Crystal Ball inNormal Speed mode.

For long-term projects, a company comprising many divisions may find thatsharing the ROV tool across divisions brings benefits in terms of better communi-cation and understanding among division managers. In particular, the benefits ofusing the ROV tool to monitor progress in a cross-divisional project include:

■ The spreadsheets become living documents that are updated continually to reflectcurrent assumptions and the prevailing business environment. If many divisionsunderstand and share the same model, discussions between divisions can be farmore productive than they otherwise might be. By discussing the assumptionsunderlying a common model, disagreements can focus on specific assumptionsin the model. This is more productive than discussions that occur sometimesin which the discussants argue about different underlying assumptions withoutrealizing that they are doing so.

■ The tool documents all assumptions to ensure consistency between decisions. Assome projects take years to develop, changing conditions in the business climatecan cause the company-wide assumptions about the conditions affecting futurecash flow to change considerably over time. The ROV tool helps to documentthe changes in these assumptions so that everyone stays ‘‘on the same page.’’

■ The modeling process itself leads to greater understanding. By decomposing theproject into its components and the relationships between them, managers seethe problem from many different aspects, which helps to gain understanding.Yet when the model is run in Steps 4 or 7 in Figure 13.2, the big picture willalso be easily seen.

■ The tool enables risk analysis of outcomes. As discussed in Chapter 7, bygenerating distributions of present value rather than a point estimate, managersgain a better idea of the riskiness of the projects they manage. Further, thedistributions allow for calculation of VaR or CVaR, as described in Chapter 10,or other measures of risk as desired in specific situations.

■ Crystal Ball enables sensitivity analysis of inputs. Sensitivity analysis can beaccomplished in several ways, including the use of the sensitivity chart to seehow each stochastic assumption affects the forecast(s), as well as an analysis ofhow the changes in the assumed parameters of the model will affect the results.This helps the managers to understand the problem better.

■ The ROV tool finds optimal solutions for specified assumptions. As with anymathematical model, its usefulness must be judged in the context of its specificapplication. OptQuest may find the optimal solution(s) for the assumptions it is

Real Options 199

fed, but there may well be non-quantifiable factors (political issues, for example)that also affect the decision. These non-quantifiable factors may cause the valuesof the decision variables chosen for implementation to be different from thevalues indicated by OptQuest, but by using it to compare expected NPV fromboth sets of decision variable values, the ROV tool will be able to provide anidea of the cost of the nonquantifiable factors.

Use of ROV Tool in New Product Development

As an example of how the ROV tool can be used at various phases throughout theproduct development process, consider the process depicted in Figure 13.3, whichis intended to represent a generic new product development project. Assume thatthere are two competing technologies available initially that can be used in theproduct.

During Phase 1 the two technologies under consideration are evaluated alongwith two market segments and three sources of costs that have some uncertainty. Thewidths of the boxes representing the technologies and markets in the tornado graphat Phase 1 are wider than the boxes for costs because the uncertainty surroundingtechnology and markets is greater at this earliest phase. The ROV model helps toquantify the uncertainties and measure their impact on expected net present value.At Decision 1, decisions about which technologies to employ are made, and some ofthe uncertainty is resolved as decision makers learn more about the project in partthrough building and revising the ROV model.

During Phase 2, the reduced uncertainty regarding the technology is depicted bythe ‘‘Tech’’ bars having smaller widths and thus moving down in the tornado graph.At this phase, most of the uncertainty is in regard to markets and operating costs.Decisions regarding the design of the product are made at Decision 2. Matchingproduct design to market opportunity is critical at this stage.

Technology 1

Technology 2

Market 1

Market 2

Market 1

Market 2

Market 1

Market 2

Market 3

Market 4

Cost 1 Tech 2

Tech 1

Cost 2

Cost 3

Phase 1Evaluate

Technology

Phase 2Design/Refine

Product

Phase 3Bring ToMarket

Decision1

Decision2

Decision3

Cost 1

Cost 1

Cost 2

Tech 3

Cost 2

Cost 3

FIGURE 13.3 Managing risk and return throughout the product development process.


Because the technology has already been been chosen at Phase 3, the great-est uncertainties surround the markets for the product during this phase. Someuncertainty remains in regard to costs and a third competing technology that hasemerged since Phase 1, but in this example, the ROV model is most useful for evalu-ating the options available for marketing the product. At Decision 3, the marketingdecisions are made.

By linking the ROV model to stochastic demand, and taking into account theuncertainty surrounding technology and operating costs, the decision makers gain abetter understanding of the impacts of these variables on their decisions. The ROVtool provides confidence bounds on its estimates, enables sensitivity analysis of itsinputs, and leads to sound business decisions based upon expected net present valueor other summary measures of interest to management.

The ROV tool is an extension of the business-case Excel models that arealready in use at most companies. Thus it can be used with existing financialmodels for strategic planning, comparing products offered by different vendors,or estimating return on capital invested. Further, by adapting models to changingbusiness conditions or decisions that have been made, the ROV tool helps tofacilitate corporate memory and fosters consistency in decision making over time.With endorsement and commitment from top management, its use adds value toexisting decision-making processes, encourages the establishment and monitoring ofmilestones for evaluating options resulting from managerial flexibility, and providesan ongoing framework within which learning from past successful and unsuccessfulprojects can be used to improve future decisions. Cooper, Edgett, and Kleinschmidt(2002) encourage managers to build in more effective go/kill decision points, andinstill a regular management review process to make these decisions. The ROV toolis of great help in this process.

SUMMARY

This chapter has provided guidelines for developing business case models using theROV tool. The tasks required include selecting inputs as stochastic assumptions,building and revising the model, adding and revising assumptions, and selectingand defining decision variables. Sensitivity analysis can be useful in identifying theassumptions that are most important for making a correct decision. The modelbuilding process is ongoing. Once a functional ROV model has been developed,additional information can be incorporated into the model as it becomes available.This helps to facilitate corporate memory, and fosters consistency in decision makingover time.

The ROV tool approach to the valuation of managerial flexibility is itself highlyflexible in its ability to support managerial decisions in a wide variety of situationsinvolving real options. The greatest benefits from using the ROV tool will come tomanagers when the tool is adopted for making decisions on a company-wide basis.Using the structured approach of the ROV tool for decision making helps to ensureconsistency in decision making and to facilitate corporate memory and learning.

Real Options 201

The ROV tool can be used for strategic planning, comparing products offeredby different vendors, or supplement the use of existing financial models for esti-mating return on invested capital. With endorsement and commitment from topmanagement, its use adds tremendous value to existing decision-making pro-cesses and provide an ongoing framework that can be used to improve futuredecisions.

APPENDIX ACrystal Ball’s Probability

Distributions

T his appendix lists a short description of each distribution in the Crystal Ball galleryalong with its probability distribution function or probability density function

(PDF), cumulative distribution function (CDF) where available, mean, standarddeviation, and typical uses. For more information about these distributions, seeEvans, Hastings, and Peacock (1993), Johnson, Kemp, and Kotz (2005), Johnson,Kotz, and Balakrishnan (1994), Law and Kelton (2000), or Pitman (1993).

All of Crystal Ball’s distributions can be truncated on either or both ends toadapt to the circumstances of your model. Truncation is accomplished by enteringthe desired values in the truncation fields. For example, in Figure A.1, the normallydistributed total return on a stock with nominal mean return 10 percent and nominal

FIGURE A.1 Normal distribution of a stock return truncated at −100 percent to reflect the limitedliability of stock ownership.

202

Crystal Ball’s Probability Distributions 203

standard deviation 50 percent is truncated at −100 percent to reflect the limitedliability of stock ownership.

When Crystal Ball truncates a distribution, the probability distribution is rescaledso that the total probability is 100% that a value will be generated within the rangedefined by the truncation points. For example, a random variable generated from thedistribution shown in Figure A.1 has a 100 percent probability of falling between−100 percent and positive infinity. Therefore, truncation will affect the actualmean and standard deviation of a random variable. In general, it is not easy todetermine the actual parameters of a truncated distribution analytically. However,you can obtain these values by selecting View → Statistics from the top menu inthe assumption’s dialog window. For example, even though the mean and standarddeviation are specified in Figure A.1 to be 10 percent and 50 percent, the actualmean and standard deviation of the random values generated by this truncateddistribution are 11.80 percent and 47.95 percent.

BERNOULLI

The Bernoulli distribution is the simplest discrete distribution. Among other uses, itrepresents the toss of a coin, if we define ‘‘1’’ to mean ‘‘heads’’ and ‘‘0’’ to mean‘‘tails’’ (or vice versa). For a fair coin, the probability, p, of obtaining heads is 0.5as depicted in Figure A.2. However, a Bernoulli trial can represent a biased (unfair)coin by specifying a different value for p. In financial modeling, it can be used tomodel the occurrence of a single event, such as the possible entry of a competitorinto your market, for example.

The Bernoulli distribution is called the yes-no distribution in Crystal Ball. Seethe yes-no section of this appendix for more details. Bernoulli assumptions can becombined to generate values from other distributions. For example: the binomialdistribution describes the number of successes in n Bernoulli trials; the geometricdistribution describes the number of failures before the first success in a sequence of

FIGURE A.2 Bernoulli distribution representing the number of heads obtained (0 or 1) with one flip ofa fair coin.


Bernoulli trials; and the negative binomial describes the number of Bernoulli trialsto get exactly β successes.

BETA

The standard beta distribution is defined for continuous values of x between 0 and1, but Crystal Ball lets you select any minimum and maximum values, then it scalesthe distribution to fit on that range with a shape determined by the alpha and betaparameters you specify. The beta distribution can represent a random proportionor probability, the time to complete a task, or as a rough model when you have nohistorical data to use with Crystal Ball’s distribution fitting routine. For much moreinformation about the beta distribution, see Gupta and Nadarajah (2004).

Parameters: Minimum, the minimum value, a; Maximum, the maximum value,b; Alpha, the first shape parameter, α > 0; Beta, the second shape parameter,β > 0. See Figure A.3 for an example of the Beta PDF with a = −10, b = 10,α = 2, and β = 3.

PDF:

f (x) =

zα−1(1 − z)β−1

B(α, β)if 0 < x − a < b − a,

0 otherwise

where z = x−ab−a , B(α, β) is the beta function, defined by B(α, β) = ∫ 1

0 tα−1(1 −t)β−1 = �(α)�(β)

�(α+β) for any real numbers α > 0, and β > 0, and �(·) is the

FIGURE A.3 Beta distribution with a = −10, b = 10, α = 2, and β = 3.


Gamma function defined by �(y) = ∫ ∞0 ty−1e−tdt for any real number y > 0.

Note that �(k + 1) = k! for any nonnegative integer k, where k! = k · (k −1) · · · (2) · (1) is read as ‘‘k-factorial.’’

CDF: No closed form.Mean:

a + α

α + β(b − a)

Standard deviation:

(b − a)

√αβ

(α + β)2(α + β + 1)

Excel function: This distribution can be defined in two ways. Use

CB.Beta(Alpha,Beta,Scale,LowCutoff,HighCutoff,NameOf)

to define beta assumptions where a = 0, and b = Scale. If the distributionhas a minimum value not equal to zero, use

CB.Beta2(Min,Max,Alpha,Beta,HighCutoff,LowCutoff,NameOf).

where Min = a, Max = b, Alpha = α, and Beta = β.Notes: The beta distribution is U shaped if α > 1 and β > 1, and is J shaped

if (α − 1)(β − 1) < 0. For all other permissible values of α and β it isunimodal.

BINOMIAL

The binomial distribution is a discrete distribution of the sum of n Bernoulli trialswith constant probability of success, p, so it represents the number of successes in aspecified number of attempts if the chance of success is the same for every attemptand the attempts are independent.

Parameters: Probability, the probability of success, p, such that 0 < p < 1;Trials, the total number of trials, n, where n is an integer such that1 ≤ n ≤ 1000. See Figure A.4 for an example of the Binomial probabilitydistribution function with p = 0.5, and n = 50.

PDF:

f (x) =

n!x!(n − x)!

px(1 − p)n−x for x = 0, 1, 2, . . . , n

0 otherwise


FIGURE A.4 Binomial(0.5,50) distribution.

CDF:

F(x) =

x∑

y=0

n!y!(n − y)!

py(1 − p)n−y for x = 0, 1, 2, . . . , n

0 otherwise

Mean:np

Standard deviation: √np(1 − p)

Excel function:

CB.Binomial(Prob,Trials,LowCutoff,HighCutoff,NameOf)

where Prob = p, and Trials = n.Notes: The binomial distribution is equivalent to the distribution of a sum of

Bernoulli random variables with the same probability of success, p. Thus,the sum of a binomial(p, n1) variable and a binomial(p, n2) variable has thebinomial(p, n1 + n2) distribution. However, the sum of binomial distribu-tions with different values of p does not follow a binomial distribution. Thebinomial distribution is symmetric when p = 0.5.You cannot specify n > 1000 in Crystal Ball. To model such a situation,use as an approximation the Normal distribution with mean and standarddeviation computed according to the expressions above, and truncated


at 0 and n + 0.99999. Use Excel’s =ROUNDOWN(number,num digits)command to obtain a discrete value, if desired.A beta binomial distribution can be simulated in Crystal Ball by definingthe parameter p in a binomial distribution as a beta random variable. Seefile AppendixA.xls.

CUSTOM

The Custom distribution is defined by specifying a list of discrete values, continuousranges of values, or discrete ranges of values, along with the associated probabilities.Once you choose the Custom from the Distribution Gallery, select Parameters fromthe top menu to specify the type of values you wish to use. You may enter the datavalues and probabilities directly in the dialog, or load them in from the worksheetby clicking the Load Data. . . button. You may also use the Excel function

CB.Custom(CellRange,NameOf)

where CellRange contains the data, and NameOf is the name of the assumption. Seefile AppendixA.xls for examples.

The custom distribution is very flexible, and is easily understood by inspectionof the following examples:

■ See Figure A.5 for an example of the custom PDF with unweighted values. Thisis specified by a list of discrete values, each of which will occur with the sameprobability.

FIGURE A.5 Custom distribution specified with Unweighted Valuesparameters.


FIGURE A.6 Custom distribution specified with Weighted Values parameters.

■ See Figure A.6 for an example of the custom PDF with weighted values. Thisis specified by a list of discrete values and their associated probabilities ofoccurrence.

■ See Figure A.7 for an example of the custom PDF with continuous ranges. Thisis specified by ranges of values within which the continuous values have equalprobability of occurrence by default.

FIGURE A.7 Custom distribution specified with Continuous Ranges parameters.


FIGURE A.8 Custom distribution specified with Discrete Ranges parameters.

■ See Figure A.8 for an example of the custom PDF with discrete ranges values.This is specified by ranges of values within which the discrete values have equalprobability of occurrence.

■ See Figure A.9 for an example of the custom PDF with sloping ranges values.This is specified by ranges of discrete values within which the probabilitiesincrease or decrease linearly.

Note that the vertical axes in Figures A.5 through A.9 are all labeled ‘‘RelativeProbability.’’ This means that the probabilities that you specify to define a customdistribution do not have to sum to 1.0; however, the specified probabilities are scaledby Crystal Ball such that the values used during the simulation do sum to 1.0.

DISCRETE UNIFORM

The discrete uniform distribution is used for modeling the random occurrence ofone of several possible outcomes, each of which is equally likely. It may be used asa first model in the absence of data for modeling a quantity that varies among theintegers {a, a + 1, a + 2, . . . , b − 1, b}, but about which little else is known.

Parameters: Minimum, the minimum value, a, an integer where −∞ < a < ∞;and Maximum, the maximum value, b, and integer where −∞ < b < ∞,and a < b. See Figure A.10 for an example of this pdf.


FIGURE A.9 Custom distribution specified with Sloping Ranges parameters.

FIGURE A.10 Discrete uniform distribution with a = 0, and b = 11.


PDF:

f (x) =

1b − a + 1

for a < x < b,

0 otherwise

where x is an integer.CDF:

F(x) =

0 if x < a,�x� − a + 1b − a + 1

for a < x < b,

1 if b < x

where �x� denotes the greatest integer less than or equal to x.Mean:

b − a2

Standard deviation: √(b − a + 1)2 − 1

12

Excel function:

CB.DiscreteUniform(Min,Max,LowCutoff,HighCutoff,

NameOf)

where Min = a, and Max = b.Notes: The discrete uniform distribution for a = 0, and b = 1 is the same as the

yes-no distribution with p = 0.5.

EXPONENTIAL

The exponential distribution is used to model continuous random variables that arenonnegative. It is used primarily to model the time between random events thatoccur at a constant average rate, such as the time between customer arrivals toservice facilities.

Parameters: Rate, the constant average rate, λ > 0. See Figure A.11 for anexample of the Exponential distribution with λ = 10.

PDF:

f (x) ={

λe−λx = for x ≥ 00 otherwise


FIGURE A.11 Exponential distribution with λ = 10.

CDF:

F(x) ={

1 − e−λx for x ≥ 00 otherwise

Mean:1/λ

Standard deviation:1/λ

Excel function:

CB.Exponential(Rate,LowCutoff,HighCutoff,NameOf)

where Rate = λ.Notes: The exponential distribution with rate λ is a special case of the gamma

distribution (with L = 0, s = λ, and β = 1), and the Weibull distribution(with L = 0, s = λ, and β = 1).The exponential is the only continuous distribution with the memorylessproperty, which for the exponential random variable, X, is defined by

Pr(X > s + t|X > s) = Pr(X > t) for all s, t > 0.

For customer arrivals occurring at a constant average rate λ, the memorylessproperty implies that no matter how long it has been since a customer


has arrived, the time until the next arrival still follows the exponentialdistribution with rate λ.Values from the Laplace, which is also known as the double exponentialdistribution, can be generated easily in Crystal Ball as follows. In one cell,define an assumption X as an exponential distribution with parameter λ,and in another cell define an assumption B as a yes-no distribution withp = 0.5. In a third cell, put in the formula Y = (2B − 1)X. Then Y followsthe Laplace distribution with mean 0 and standard deviation 1/

√λ.

GAMMA

The gamma is a continuous distribution used often for modeling the time requiredto complete some task, such as repairing a machine or waiting on a customer in aservice facility.

Parameters: Location, the location parameter, L; Scale, the scale parameter,s > 0; and Shape, the shape parameter, β > 0. See Figure A.12 for anexample of the beta distribution with L = 0, s = 1, and β = 2.

PDF:

f (x) =

(x−L

s

)β−1e− x−L

s

�(β)sfor x > L

0 otherwise

FIGURE A.12 Gamma distribution with L = 0, s = 1, and β = 2.


where �(·) is the gamma function defined in the beta distribution section ofthis appendix.

CDF: If β is not an integer, there is no closed form; for β an integer

F(x) =

1 − e− x−L

s∑β−1

i=0

(x−L

s

)i

i!for x > L

0 otherwise

Mean:sβ

Standard deviation:s√

β

Excel function:

CB.Gamma(Loc,Scale,Shape,LowCutoff,HighCutoff,NameOf)

Notes: The gamma distribution with L = 0, and β = 1 is the same as theExponential distribution with rate s.The gamma distribution with L = 0, and β = k, where k is a positive integeris called the k-Erlang distribution with rate s.For a positive integer, k, the gamma distribution with L = 0, β = k/2, ands = 2 is the same as the chi-square distribution with k degrees of freedom.

GEOMETRIC

The geometric distribution is used to model the number of trials to get the firstsuccess in a sequence of IID Bernoulli trials with probability p of success on eachtrial. For example, it can be applied to model the number of calls a salespersonmakes to obtain her first sale, the number of items in a batch of random size, or thenumber of items demanded by a customer.

Parameters: Probability, the probability of success, p. See Figure A.13 for anexample of the geometric PDF with p = 0.2.

PDF:

f (x) ={

p(1 − p)x−1 for x ∈ {1, 2, 3, . . .0 otherwise

CDF:

F(x) ={

1 − (1 − p)�x� for x ≥ 10 otherwise


FIGURE A.13 Geometric distribution with p = 0.2.

Mean:1p

Standard deviation: √1 − pp

Excel function:

CB.Geometric(Prob,LowCutoff,HighCutoff,NameOf)

where Prob = p.Notes: The geometric distribution is a discrete analogue of the exponential

distribution, and is the only discrete distribution with the memorylessproperty, which for the geometric random variable, Y, is defined by

Pr(Y − k = m|Y ≥ k) = Pr(Y = m) for k ≥ 0 and m = 0, 1, . . .

For a gambler making the same bet on roulette that has probability p ofwinning, the memoryless property implies that no matter how many timesthe gambler has bet, the number of spins of the roulette wheel until thegambler wins still follows the geometric distribution.


An alternative form of the geometric distribution involves the number ofBernoulli trials up to, but not including, the first success. The randomvariable defined by a draw from a geometric distribution with probability,p, follows the negative binomial distribution with probability, p, and shapeparameter, β = 1.

HYPERGEOMETRIC

The hypergeometric is the discrete distribution of the number of successes in a sampledrawn without replacement from a population with known numbers of successesand failures.

Parameters: Success, the number of successful items in the population, Nx;Trials, the number of items in the sample, n; and Population, the populationsize, N. The number of failures in the population is N − Nx. See Figure A.14for an example of the hypergeometric distribution with Nx = 50, n = 50,and N = 100.

PDF:

f (x) =

(Nxx

)(N−Nxn−x

)(N

n

) for a ≤ x ≤ b

0 otherwise

where x is an integer, a = max[0, n − N + Nx], and b = min[Nx, n].

FIGURE A.14 Hypergeometric distribution with Nx = 50, n = 50, and N = 100.


CDF:

F(x) =

0 for a < x�x�∑

i=0

(Nxi

)(N−Nxn−i

)(N

n

) for a ≤ x ≤ b

1 for b < x

Mean:

n(

Nx

N

)

Standard deviation: √

n(

Nx

N

) (1 − Nx

N

) (N − nN − 1

)

Excel function:

CB.Hypergeometric2(Success,Trials,Population,LowCutoff,HighCutoff,

NameOf)

where Success = Nx, Trials = n, and Population = N.Notes: You cannot specify N > 1000 or n > 1000 for Crystal Ball’s hypergeo-

metric distribution. To model such a situation, use as an approximation thenormal distribution with mean and standard deviation computed accordingto the expressions above, and truncated at 0 and n + 0.99999. Use Excel’s=ROUNDDOWN(number,num digits) command to obtain a discrete value.

LOGISTIC

The logistic is a continuous distribution that appears often near the top of the listof distributions Crystal Ball suggests when fitting distributions to stock returns andother financial data. It has fatter tails than the normal distribution. The excesskurtosis of the logistic distribution is 1.2. The logistic distribution has been appliedto models in the areas of population growth, bioassay, medical diagnosis, andpublic health, among others. For much more about the logistic distribution, seeBalakrishnan (1991).

Parameters: Mean, the mean of the distribution, µ; and Scale, the scaleparameter, s > 0. See Figure A.15 for an example of the standard logisticdistribution, which has µ = 0 and s = 1.

PDF:

f (x) =

sech2

[x − µ

2s

]

4sfor − ∞ < x < ∞


FIGURE A.15 Logistic distribution with µ = 0 and s = 1.

or

f (x) ={

zs(1 + z)2

for − ∞ < x < ∞

where z = e−(x−µ)/s

CDF:

F(x) ={

1 − 11 + z

for − ∞ < x < ∞

Mean:µ

Standard deviation:πs√

3

Excel function:

CB.Logistic(Mean,Scale,LowCutoff,HighCutoff,NameOf)

where Mean = µ and Scale = s.Notes: Because the PDF of the logistic distribution can be expressed in terms of

the square of the hyperbolic secant function, sech, it is sometimes called thesech-squared distribution.


LOGNORMAL

If ln(X), the natural logarithm of random variable X, follows a normal distribution,then X is said to follow the lognormal distribution. The lognormal distributionarises often in financial modeling and risk analysis because of the product version ofthe central limit effect (see section 4.1.7).

Parameters: Mean, the mean, µL > 0; and Std. Dev., the standard deviation,σL > 0. See Figure A.16 for an example of the lognormal PDF with µL =2.72 and σL = 1.

PDF:

f (x) =

1

x√

2πσ 2e−(ln x−µ)2/2σ2

for x > 0

0 otherwise

where

µ = ln µL − 12

ln(

1 + σ 2L

µ2L

)

and

σ =√

ln(

1 + σ 2L

µ2L

)

FIGURE A.16 Lognormal distribution with µL = 2.72 and σL = 1.



µL = eµ+σ2/2

Standard deviation:σL =

√e2µ+σ2

(eσ2 − 1

)

Excel function:

CB.Lognormal2(LogMean,LogStdDev,LowCutoff,HighCutoff,NameOf)

where LogMean = µL, and LogStdDev = σL. You may also use

CB.Lognormal(Mean,StdDev,LowCutoff,HighCutoff,NameOf)

where Mean = µ, and StdDev = σ .Notes: Be sure to keep clear the difference between µL and σL, the mean and

standard deviation of the lognormal random variable, X; and µ and σ , themean and standard deviation of the normal distribution followed by ln(X),the natural logarithm of X.

MAXIMUM EXTREME

The maximum extreme distribution is the positively skewed form of the extremevalue distribution. Crystal Ball’s maximum extreme distribution is sometimes calledthe type 1 extreme value distribution. It has been applied to models in the areas offlood flows, radioactive emissions and human lifetimes, rupture of solids, earthquakemagnitudes, estimation of insurance premiums, and stock market movements, amongothers. For more information, see de Haan and Ferreira (2006).

Parameters: Likeliest, the mode, m; and Scale, the scale parameter, s > 0. SeeFigure A.17 for an example of this PDF with m = 0 and s = 1, which is calledthe standard maximum extreme distribution, or the Gumbel distribution.

PDF:

f (x) ={ z

se−z for − ∞ < x < ∞0 otherwise

where z = e−(x−m)/s.CDF:

F(x) ={

e−z for − ∞ < x < ∞0 otherwise


FIGURE A.17 Maximum extreme distribution with m = 0, and s = 1.

Mean:m + 0.57722s

Standard deviation: sπ√6

Excel function:

CB.MaxExtreme(Likeliest,Scale,LowCutoff,HighCutoff,NameOf)

where Likeliest = m, and Scale = s.Notes: The maximum extreme distribution has skewness coefficient 1.139547,

and excess kurtosis 2.4.Because of the functional form of f (x), the maximum extreme distributionis sometimes called the doubly exponential distribution. Do not confuse thedoubly exponential distribution with the double exponential (aka Laplace)distribution.

MINIMUM EXTREME

Parameters: Likeliest, the mode, m; and Scale, the scale parameter, s > 0. SeeFigure A.18 for an example of this PDF with m = 0, and s = 1, which iscalled the standard minimum extreme distribution.


FIGURE A.18 Minimum extreme distribution with m = 0, and s = 1.

PDF:

f (x) ={ z

se−z for − ∞ < x < ∞0 otherwise

where z = e(x−m)

s .CDF:

F(x) ={

e−z for x ≥ 00 otherwise

Mean:m − 0.57722s

Standard deviation: sπ√6

Excel function:

CB.MinExtreme(Likeliest,Scale,LowCutoff,HighCutoff,NameOf)

where Likeliest = m and Scale = s.Notes: The minimum extreme distribution has skewness coefficient −1.139547,

and excess kurtosis 2.4.


NEGATIVE BINOMIAL

The negative binomial assumption in Crystal Ball is the discrete distribution ofthe total number of Bernoulli trials required to get exactly β successes where eachBernoulli trial has probability of success, p. Thus, the smallest value that a CrystalBall negative binomial assumption can generate is β, and the largest potentialnumber is infinitely large.

Parameters: Probability, the probability of success on each trial, p where 0 <

p < 1; and Shape, the number of successes, β, where β > 0 is an integer.See Figure A.19 for an example of the negative binomial PDF with p = 0.2,and β = 10.

PDF:

f (x) ={ (x−1

β−1

)pβ(1 − p)x−β for x ≥ β

0 otherwise

where x is an integer.CDF:

F(x) =

�x�∑

i=β

(i − 1β − 1

)pβ(1 − p)i−β for x ≥ β

0 otherwise

FIGURE A.19 Negative binomial distribution with p = 0.2, and β = 10.


Mean:β/p

Standard Deviation: √β(1 − p)/p

Excel Function:

CB.NegBinomial(Prob,Shape,LowCutoff,HighCutoff,NameOf)

where Prob = p and Shape = β.Notes: The negative binomial distribution is also defined for noninteger β, but

is not implemented in Crystal Ball for such values of β.

NORMAL

The normal is arguably the best known continuous distribution because of theCentral Limit Theorem (see Chapter 4) and its application in many fields. TheNormal is sometimes called the Gaussian distribution. For more information aboutthe Normal distribution, see Patel and Read (1996).

Parameters: Mean, the location parameter, µ, where −∞ < µ < ∞; andStd. Dev., the scale parameter, σ , where σ > 0. See Figure A.20 for anexample of the standard normal distribution, which has parameters µ = 0,and σ = 1.

FIGURE A.20 Normal distribution with µ = 0 and σ = 1.


PDF:

f (x) =

1√2πσ

e−(x−µ)2/2σ2for − ∞ < x < ∞

0 otherwise


µ

Standard deviation:σ

Excel function:

CB.Normal(Mean,StdDev,LowCutoff,HighCutoff,NameOf)

where Mean = µ and StdDev = σ .Notes: The normal distribution is symmetric, so has skewness coefficient 0.

The kurtosis coefficient of any normal distribution is 3. Because the normaldistribution is the standard to which the kurtosis coefficient of any otherdistribution is often compared, statisticians have defined the excess kurtosiscoefficient to be equal to the kurtosis coefficient minus 3.

PARETO

The Pareto is a continuous distribution first used by economist Vilfredo Pareto inthe late 1800s to describe the distribution of income over a population. For moreinformation about the Pareto distribution, see Arnold (1983).

Parameters: Location, the location parameter, L, where L > 0; and Shape, theshape parameter, β, where β > 0. See Figure A.21 for an example of thePareto distribution with L = 1, and β = 2.

PDF:

f (x) ={

βLβ/xβ+1 for x ≥ L0 otherwise

CDF:

F(x) ={

1 − (L/x)β for x ≥ L

0 otherwise

Mean:βL

β − 1for β > 1

Standard deviation:βL2

(β − 1)2(β − 2)for β > 2


FIGURE A.21 Pareto distribution with L = 1 and β = 2.

Excel function:

CB.Pareto(Loc,Shape,LowCutoff,HighCutoff,NameOf)

where Loc = L, and Shape = β.Notes: Pareto’s work gave rise to the so-called 80–20 rule, whereby it is

maintained that 80 percent of the wealth of a society is owned by 20 percentof the population. This rule has been expanded to other applications, andforms the basis of Pareto charts in Six Sigma quality management.

POISSON

The Poisson is the discrete distribution of the number of events that occur in a fixedarea of opportunity when the events are occurring at a constant rate.

Parameters: Rate, the constant rate of occurrence, λ, where λ > 0. SeeFigure A.22 for an example of the Poisson distribution with λ = 10.

PDF:

f (x) ={

λxe−λ/x! for x ≥ 00 otherwise

where x is an integer.


FIGURE A.22 Poisson distribution with λ = 10.

CDF:

F(x) =

x∑

i=0

λie−λ/i! for x ≥ 0

0 otherwise

Mean:λ

Standard deviation: √λ

Excel function:

CB.Poisson(Rate,LowCutoff,HighCutoff,NameOf)

where Rate = λ,Notes: The Poisson distribution is usually applied to situations in which the

number of potential opportunities for the outcomes to occur is large, butthe probability of each occurrence is relatively small.A notable application of the Poisson distribution was the number of deathsper year from kicks by horses in the Prussian Army Corps (Bortkiewicz1898; see also Quine and Seneta 1987).


STUDENT’S T

Parameters: Midpoint, the midpoint of the distribution, m, where −∞ < m <

∞; Scale, the scale parameter, s, where s > 0; and Deg. Freedom, the num-ber of degrees of freedom, d, an integer where 0 < d ≤ 30. See Figure A.23for an example of the Student’s t distribution with m = 0, s = 1, and d = 5.

PDF:

f (x) =

�

(d + 1

2

)

√dπ�

(d2

)(1 + z2

d

) d+12

for − ∞ < x < ∞

0 otherwise

where z = x−ms , and �(·) is the Gamma function defined in the discussion of

the beta distribution in this appendix.CDF:

F(x) =

12

+ tan−1

(z√d

)+ z

√d

d + z2×

(d−3)/2∑

j=0

aj(1 + z2/d

)j

for d odd and − ∞ < x < ∞12

+ z

2√

d + z2×

(d−2)/2∑

j=0

bj(1 + z2/d

)j

for d even and − ∞ < x < ∞0 otherwise

FIGURE A.23 Student’s t distribution with m = 0, s = 1, and d = 5.


where z = x − ms

, aj = [2j/(2j + 1)] aj−1, a0 = 1, bj = [(2j − 1)/2j] bj−1, and

b0 = 1.Mean:

m, for d > 1

Standard deviation: √sd/(d − 2), for d > 2

Excel function:

CB.StudentT(Midpoint,Scale,Degrees,LowCutoff,HighCutoff,NameOf)

Notes: The Student’s t distribution with m = 0 and d = 1 is called the standardCauchy distribution, for which the mean and standard deviation do notexist (are infinite).

TRIANGULAR

The triangular distribution is used most often as a rough distribution in the absenceof data. Its use as an approximation to the normal distribution is discussed inBell (1962).

Parameters: Minimum, the minimum value, a, where −∞ < a < ∞; Likeliest,the mode, m, where −∞ < a ≤ m < ∞; and Maximum, the maximumvalue, b, where −∞ < a ≤ m ≤ b < ∞, but a < b. See Figure A.24 for anexample of the Triangular distribution with a = −10, m = 0, and b = 10.

FIGURE A.24 Triangular distribution with a = −10, m = 0, and b = 10.


PDF:

f (x) =

2(x − a)(m − a)(b − a)

for a ≤ x ≤ m

2(b − x)(b − m)(b − a)

for m < x ≤ b

0 otherwise

CDF:

F(x) =

0 for x < a(x − a)2

(m − a)(b − a)for a ≤ x ≤ m

1 − (b − x)2

(b − m)(b − a)for m < x ≤ b

1 for b < x

Mean:a + m + b

3

Standard deviation: √a2 + m2 + b2 − am − ab − bm

18

Excel function:

CB.Triangular(Minimum,Likeliest,Maximum,LowCutoff,HighCutoff,

NameOf)

where Minimum = a, Likeliest = m, and Maximum = b.Notes: The standard triangular distribution is obtained when a = 0 and b = 1.

If m = 1/2, the standard triangular distribution is symmetric. For moreinformation about the triangular distribution, see Ayyangar (1941).

UNIFORM

The uniform distribution is used most often as a rough distribution in the absenceof data. It applies to any continuous random variable for which the largest andsmallest possible values can be specified, with equal likelihood for the occurrence ofany value in between the minimum and maximum values.

Parameters: Minimum, the minimum value, a, where −∞ < a < ∞; andMaximum, the maximum value, b, where −∞ < a < b < ∞. SeeFigure A.25 for an example of the uniform distribution with a = −10and b = 10.


FIGURE A.25 Uniform distribution with a = −10 and b = 10.

PDF:

f (x) =

1b − a

for a < x < b

0 otherwise

CDF:

F(x) =

0 for x < ax − ab − a

for a ≥ x ≥ b

1 for b < x

Mean:a + b

2

Standard deviation:b − a√

12

Excel function:

CB.Uniform(Min,Max,LowCutoff,HighCutoff,NameOf)

where Min = a, and Max = b.Notes: The uniform distribution with a = 0 and b = 1 is used in generating

variates from all other distributions in Crystal Ball. See Appendix B fordetails.


WEIBULL

The Weibull distribution is widely used in engineering practice to represent thelifetime of a system component composed of many parts that fails when the firstof these parts fails. The Weibull has been applied to problems in the areas of tidalheight, efficacy of medical treatment, insurance claims, maintenance of street lights,rock blasting, and spare part planning. For more about the Weibull distribution, seePrabhakar Murthy, Min, and Jiang (2004).

Parameters: Location, the location parameter, L, where −∞ < L < ∞; Scale,the scale parameter, s, where s > 0; and Shape, the shape parameter, β,where β > 0. See Figure A.26 for an example of the Weibull distributionwith L = 0, s = 1, and β = 2.

PDF:

f (x) =

β

s

(x − L

s

)β−1

e−

x − L

s

for x ≥ L

0 otherwise

CDF:

F(x) =

1 − e−

x − L

s

β

for x ≥ L0 otherwise

FIGURE A.26 Weibull distribution with L = 0, s = 1, and β = 2.


Mean:

L + s�[β + 1

β

]

where �[·] is the gamma function defined in section A.2.Standard deviation:

s

√√√√(

�

[β + 2

β

]−

{�

[β + 1

β

]}2)

Excel function:

CB.Weibull(Loc,Scale,Shape,LowCutoff,HighCutoff,NameOf)

where Loc = L, Scale = s, and Shape = β.Notes: The Weibull distribution with β = 2 is also known as the Rayleigh

distribution.

YES-NO

The yes-no distribution describes a random occurrence with two possible outcomes,which are usually denoted by x = 1 (a ‘‘success’’) and x = 0 (‘‘failure’’).

Parameters: Probability of Yes(1), the probability of a success, p, where 0 < p <

1. See Figure A.27 for an example of the yes-no distribution with p = 0.8.

FIGURE A.27 Yes-no distribution with p = 0.8.


PDF:

f (x) =

1 − p for x = 0p for x = 10 otherwise

CDF:

F(x) =

0 for x < 01 − p for 0 ≤ x ≤ 1

1 for 1 ≥ x

Mean:p

Standard deviation: √p(1 − p)

Excel function:

CB.YesNo(Prob,LowCutoff,HighCutoff,NameOf)

where Prob = p.Notes: The yes-no distribution is also known as the Bernoulli distribution, and

is equivalent to the binomial distribution with n = 1.

APPENDIX BGenerating Assumption Values

This appendix provides a brief description of how Crystal Ball generates values fromits assumptions, to which we refer in this appendix as random numbers and randomvariates. For our purposes, a random number is a computer-generated number thatappears to be drawn from the uniform probability distribution with a = 0 and b = 1(see Appendix A), independently of the other random numbers. A random variateis a computer-generated number that appears to be drawn independently from anyother probability distribution.

Crystal Ball uses a variety of methods to generate independent random variatesby transforming random numbers. The choice of transformation depends on severalfactors, such as the type and parameters of the distribution, as well as the samplingmethod used (Monte Carlo or Latin Hypercube). For illustration, the inversetransform random-variate generation method is described below, and referencesare given for the other methods Crystal Ball uses. For more information aboutgenerating random numbers and variates, see Bratley, Fox, and Schrage (1987),Fishman (2006), Glasserman (2004), Law and Kelton (2000), and the referenceslisted in the Bibliography section of the Crystal Ball User Manual.

GENERATING RANDOM NUMBERS

Crystal Ball generates random numbers using a linear congruential generator (LCG).An LCG is defined by the recursive formula

Zi = (aZi−1 + c)(mod m) (B.1)

where m is the modulus, a is the multiplier, c is the increment, and Z0 is the seed.The operator ‘‘mod’’ in Expression B.1 represents modulo reduction, which meansto take the remainder after dividing by the modulus. For example, 31(mod 8) = 7,because 8 goes into 31 three times with a remainder of 7. Likewise, 3(mod 8) = 3,11(mod 8) = 3, and 890(mod 8) = 2.

Modulo reduction is used simply to ensure that the LCG delivers a sequence ofintegers, Zi such that 0 ≤ Zi ≤ m − 1. After dividing each integer by the modulus, m,we get a sequence Ui = Zi/m for i = 1, 2, . . . , m − 1 that appears to be iid U(0, 1). If

235


c > 0, the recursion in Expression B.1 is called a mixed LCG. If c = 0, the recursionis called a multiplicative LCG.

Example LCG

To understand how LCGs work, consider the mixed LCG with a = 5, c = 1, m = 8,and Z0 = 3,

Zi = (5Zi−1 + 1)(mod 8),

which is in cells A3:C23 of file LCGExamples.xls shown Figure B.1. This simpleLCG has three desirable features:

1. Because the LCG generates the m = 8 integers {0, 1, 2, 3, 4, 5, 6, 7} before itrepeats, the LCG is said to be full period.

2. The integers appear to be generated independently of each other in the sensethat they do not occur in any readily identifiable systematic order, such as {7, 6,5, 4, 3, 2, 1, 0}.

3. By specifying the seed, you can determine the rest of the stream of ‘‘random’’numbers.

FIGURE B.1 LCG defined byZi = (5Zi−1 + 1)(mod 8) with Z0 = 3.

Generating Assumption Values 237

Because the stream of numbers appears random, but is actually deterministic and inthe systematic order defined by the recursion, the values generated by an LCG areoften called pseudorandom numbers.

The parameters must be chosen carefully to give an LCG full period. Forexample, make the LCG in Figure B.1 multiplicative by setting the increment,c = 0, and the LCG will generate the set of integers {3, 7, 3, 7, . . . }, whichmakes the LCG less than full period. This is undesirable because it is inefficient,and can be misleading. If you use m − 1 random numbers from a mixed LCGthat is less than full cycle, you will be using some of the random numbers morethan once, so the estimates you obtain from the simulation will not be as preciseas they would be based on m − 1 random numbers from a mixed LCG that isfull cycle.

Now consider the multiplicative LCG with a = 5, m = 7, and Z0 = 3,

Zi = 5Zi−1 mod 7,

which is in cells A27:C47 of file LCGExample.xls shown Figure B.2. This mul-tiplicative LCG generates the m − 1 = 6 integers 1 through 6. Clearly, if amultiplicative LCG somehow generated Zi = 0 for any i, then all other Zis would

FIGURE B.2 LCG defined by Zi = 5Zi−1

mod 8 with Z0 = 3.


equal zero, too. Therefore, the multiplicative LCGs of most use are those thatgenerate the integers 1 through m − 1, where m is a large number. Fortunately,researchers have developed LCGs with good statistical properties, and Crystal Balluses one of these good LCGs in its algorithms.

Crystal Ball’s LCG

Crystal Ball version 7.2 uses the prime modulus multiplicative linear congruentialgenerator defined by

Zi = (62089911Zi−1)(mod 231 − 1),

where the modulus m = 231 − 1 = 2,147,483,647 is a prime number (i.e., evenlydivisible only by itself and 1) chosen in conjunction with a = 62089911 to givethe LCG good statistical properties and compatibility with personal computerarchitectures. For more information about choosing the parameters of an LCG, seeKnuth (1998).

Crystal Ball’s generator has period m − 1 = 231 − 2 = 2,147,483,646, so iscapable of delivering more than 2 billion distinct random numbers. By setting Seedin Run Prefs. . . you are determining Z0. If Seed is set to 0, CB will determinethe seed based on the number of milliseconds since Windows was last started onyour computer.

Excel’s Random Number Generator

Excel has a random number generator that it uses for its =rand() function. The firstrandom number is:

ri = fractional part of (9821 ∗ ri−1 + 0.211327) for i = 1, 2, . . .

where r0 = .5. Excel’s random number generator will provide up to 1 milliondifferent numbers. Through a specification in an initialization file, you can haveExcel determine its seed, r0, from the system clock. See

http://support.microsoft.com/kb/q86523/

for more information.Besides being able to generate only a small fraction of the number of random

numbers Crystal Ball can produce, this generator has relatively poor statisticalproperties. For more information about random number generators and theirstatistical properties, see the Web site,

http://random.mat.sbg.ac.at/links/rando.html.


GENERATING RANDOM VARIATES

Crystal Ball uses random numbers in a variety of methods to generate randomvariates. The particular method used depends on the assumption’s distributionfunction, and the type of sampling (Monte Carlo or Latin Hypercube) you havespecified in the Run Preferences. The specific method used to generate randomvariates from each assumption is listed in Chapter 2 of the Crystal Ball ReferenceManual. In this appendix, we will consider just one of the methods, inversetransformation, and see how Latin Hypercube sampling is used with it to makethe sampling ‘‘more even.’’

Inverse-Transform Method

Suppose that you want to generate a random variate, x, from a distribution withCDF, F(x). The inverse-transform method takes a random number, u, from theinterval (0, 1), and transforms it into a random variate, x, by using the inverse ofX’s distribution function, F−1(x).

For example, suppose X has the right triangular distribution with PDF

f (x) = x/50 for 0 ≤ x ≤ 10,

shown in Figure B.3, and CDF

F(x) = x2/100 for 0 ≤ x ≤ 10,

shown in Figure B.4. This distribution is the same as a Crystal Ball triangularassumption with a = 0, b = 10, and m = 10 (see Appendix A), so we will designate

FIGURE B.3 Triangular PDF, f (x) = x/50 for 0 ≤ x ≤ 10.


FIGURE B.4 Triangular CDF, F(x) = x2/100 for 0 ≤ x ≤ 10. This is the cumulative distributionfunction for the PDF given in Figure B.3. The figure also illustrates the inverse-transform algorithm. ForU = .16, F−1(.16) = 10

√.16 = 4, and for U = .81, F−1(.81) = 10

√.81 = 9.

it as triangular(0,10,10). Given a value, u, generated by Crystal Ball’s LCG, acorresponding value from the distribution of X is found by inverting the CDF asfollows to get an expression for x as a function of u:

F(x) = u = x2/100 ⇒ x = 10√

u.

Figure B.4 illustrates the inverse transformation. For the value u = .81, thecorresponding variate from the triangular(0,10,10) distribution is determined asx = 10

√.81 = 9; and for u = .16, the triangular(0,10,10) variate is x = 10

√.16 =

4. By generating many such values of u randomly between 0 and 1, we get aMonte Carlo sample from the triangular(0,10,10) distribution of X. See the fileInverseTransform.xls for a demonstration.

The inverse-transform method also works for discrete random variables. In somecases, it is difficult or impossible to find a closed-form formula for F−1(x), so theinverse-transform method cannot be used. However, it is also possible to generaterandom variates from the distributions for which Excel provides inverse distributionfunctions. For example, the file InverseTransform.xls shows how a uniform(0,1)random number in cell A9 is used to generate random variates from the betadistribution with parameters α = 2, β = 3, a = 0, and b = 10 (see section A.2). Thisis done by using in cell B9 the formula =BETAINV(A9,2,3,0,10). Excel’s BETAINVfunction uses a numerical approximation to the inverse CDF, F−1(x), for the betadistribution. For more information about the inverse-transform method and otherways to generate random variates, see Chapter 8 of Law and Kelton (2000), orChapter 3 of Fishman (2006).


LATIN HYPERCUBE SAMPLING

Latin Hypercube sampling (LHS) is a form of stratified sampling that Crystal Ball’sauthors have implemented to help ensure that all portions of the distribution areused to generate random variates, especially the tails. To illustrate how LHS works,consider Figure B.5, in which the triangular(0,10,10) CDF is stratified into five equalpieces by values of F(x). Stratum A includes values of F(x) between 0 and 0.2,stratum B includes values of F(x) between 0.2 and 0.4, stratum C includes values ofF(x) between 0.4 and 0.6, stratum D includes values of F(x) between 0.6 and 0.8,and Stratum E includes values of F(x) between 0.8 and 1.0. Figure B.6 shows thePDF for the triangular(0,10,10) distribution with the same strata, A through E. Byconstruction, each of the five pieces of the PDF has area 0.2.

With Latin Hypercube sampling enabled, Crystal Ball will randomly select oneof the strata, then generate a random variate, u, within that stratum and compute thecorresponding random variate x. On the next trial, it will randomly select a stratumthat has not yet been selected, and compute the random variate x corresponding tothe value of u selected. It will continue this until all five strata have been used, thenwill repeat the process until one of the stopping criteria is met.

If more trials are specified than the number of strata, then Crystal Ball willattempt to split up evenly the number of trials for which each stratum is sampled.This is why it is best in Run Preferences to make Number of trials to run on theTrials tab an integer multiple of Sample size on the Sampling tab. The value ofSample size is the number of strata into which the distribution is divided.

What is a Latin Hypercube? A Latin square has the property that each of the threesymbols A, B, and C, appear only once in each row and column of a two-dimensional

FIGURE B.5 Triangular(0,10,10) CDF, F(x) = x2/100, for 0 ≤ x ≤ 10, stratified into five equal piecesby values of F(x). The defining values are: F−1(0.2) = 4.47, F−1(0.4) = 6.32, F−1(0.6) = 7.75, andF−1(0.8) = 8.94.


FIGURE B.6 Triangular PDF stratified into five equal pieces by area. The defining values are:F−1(0.2) = 4.47, F−1(0.4) = 6.32, F−1(0.6) = 7.75, and F−1(0.8) = 8.94.

matrix, as shown below.

A B CB C AC A B

A Latin cube has the property that each symbol appears only once in each rowor column of a three-dimensional matrix, as shown below in three layers of Latinsquares.

A B CB C AC A B

C A BA B CB C A

B C AC A BA B C

A Latin Hypercube has the property that each symbol appears only once in eachrow or column of a higher-than-three-dimensional matrix.

APPENDIX CVariance Reduction Techniques

A s we saw in Chapter 12, one of the many uses of Monte Carlo simulation byfinancial analysts is to place a value on financial derivatives. You now know that

you can sharpen the point estimate of the derivative’s value by using the brute forcemethod of increasing the number of trials run during a simulation. However, thereare also other relatively simple changes that you can make to a model that providedramatic increases in precision for a given number of simulation trials. Such changesmade to a model are called variance reduction techniques. This appendix, based onCharnes (2002), shows how the variance reduction techniques of antithetic variates(AV) and control variates (CV) can be used to sharpen the precision of your estimateof the value of an Asian call option.

The best point estimate of the value of a derivative is usually the mean, orarithmetic average of the derivative’s discounted payoff taken over all trials of thesimulation. One measure of the sharpness of the point estimate of the mean isMean Standard Error, defined as

Mean Standard Error =√

VarianceTrials

. (C.1)

The precision of the mean as a point estimate is often defined as the half-width of a95% confidence interval, which is calculated as

Precision = 1.96 × Mean Standard Error. (C.2)

Lower values of Precision in Expression (C.2) correspond to sharper estimates.Increasing the number of trials is a brute force method of obtaining sharper

estimates. This reduces the Mean Standard Error by increasing the value of Trialsin the denominator of Expression(C.1). However, highly precise estimates with thebrute force method can take a long time to achieve. So-called variance reductiontechniques reduce Mean Standard Error by decreasing Variance in the numerator ofExpression (C.1) and can be used to speed up simulations by achieving a specifiedlevel of precision with a smaller number of Trials.

In this appendix, we consider two variance reduction techniques, the method ofantithetic variates and the method of control variates. The AV method is more widelyapplicable than CV, but when it can be used, the CV method is much more efficient.

243


The use of Monte Carlo simulation in pricing options was first published by Boyle(1977), but recently the literature in this area has grown rapidly. You can learn moreabout the use of variance reduction techniques from Fishman (2006), or Law and Kel-ton (2000). For a good discussion of variance reduction techniques applied to finan-cial derivatives, see Boyle, Broadie, and Glasserman (1997), or Glasserman (2004).

USING CRYSTAL BALL TO VALUE AN ASIAN OPTION

As described in Chapter 12, an Asian option is also called an average option becauseits price is linked to the average value of the underlying asset on specific dates.Suppose the prices of the underlying asset are denoted by St, for t ∈ {0, 1, . . . , T},where T is the expiration date of the option. The agreed amount for which theunderlying is traded is called the strike price, denoted by K. An average-price Asianoption has a payoff at time T based on the difference between the strike priceand the arithmetic average price of the underlying asset. Specifically, the payoff isV = max(A − K, 0), where A = ∑T

t=1 St/T. Financial analysts are often interested indetermining the value of the option, denoted here as CA.

In the Black-Scholes worldview, a fair value for an option is the present value ofthe expected value of the option payoff at expiration under a risk-neutral randomwalk for the underlying asset prices. Therefore, the formula used in a Crystal Ballmodel to generate daily asset prices on the ith trial of the simulation is

S(i)t+1 = St exp

((r − σ 2/2)(1/252) + σ

√1/252Z(i)

), (C.3)

for t = 1, 2, . . . , T, where r is the risk-free rate of interest, σ is the annual volatilityof the asset prices, and Z(i) is a standard normal random variate. Expression (C.3)assumes that there are 252 trading days in a year.

The general approach to using Crystal Ball to find the price of an average-priceAsian option is straightforward (see the file AsianCallVarReduction.xls for specificdetails):

1. For each simulation trial, i = 1, 2, . . . , N, simulate sample paths of the underlyingasset prices, St, for t = 0, 1, . . . , T according to Expression(C.3).

2. Calculate the average price, A, of the underlying asset and payoff of the optionat time T as V(i) = max(A − K, 0).

3. Compute the present value of the cash flows of the option on each sample path,as C(i)

A = V(i)e−rT/252.4. Use the average of the present values over the sample paths,

CA =N∑

i=1

C(i)A /N,

as the point estimate of the option’s value, and use the variance of the distributionof the values C(i)

A to obtain the precision of CA with Expression (C.2).

Variance Reduction Techniques 245

Crystal Ball takes care of the housekeeping details in the steps above, so that inpractice all we need to do after running the simulation model for a given number oftrials is look at the Crystal Ball Forecast window statistics to obtain the forecast cellmean, which is CA, and the mean standard error, which is a measure of precision.

ANTITHETIC VARIATES

The method of antithetic variates for variance reduction is based on the fact that ifZ(i) has a standard normal distribution, then so does −Z(i). Therefore, if we replaceZ(i) in (C.3) with −Z(i), we also get a valid sample from the distribution of stockprices at time T. In using antithetic variates with the procedure given above, weconstruct two intermediate estimates in Step 3, C+

A(Z(i)) and C−A(−Z(i)), then a final

estimate, CAVA = (C+

A + C−A)/2 as the point estimate in Step 4. Because of the way

we use Z(i) and −Z(i), the estimates C+A and C−

A both have the same expected value;however, because the two estimates are negatively correlated, the distribution of CAV

A

has a lower variance than the variance of either estimate by itself. Thus, antitheticvariates gives an estimate that has the expected value we are trying to find, but witha smaller Mean Standard Error than the estimate obtained without using a variancereduction technique.

CONTROL VARIATES

The method of control variates replaces the evaluation of an unknown expected valuewith the evaluation of the difference between the unknown quantity and a relatedquantity whose expected value is known. Here, the unknown quantity of interestis the value, CA, of an average-price Asian call option whose payoff at expirationis max (A − K, 0), where A is the arithmetic average of the underlying asset pricesduring the holding period. The related quantity with known expectation is the value,

CG, of an Asian option whose payoff is max (G − K, 0), where G =(∏T

t=1 St

)1/T

is the

geometric average. Because of the lognormality of the stock price model, an analyticexpression is available for CG, but not for CA (see Kemna and Vorst 1990 for details).

The values of interest here are denoted as CA = E[C(i)

A

], and CG = E

[C(i)

G

],

where C(i)A and C(i)

G are the discounted option payoffs for a single simulated pathof the underlying for options that pay off on the arithmetic and geometric means,respectively. Because

CA = CG + E[CA − CG

],

an unbiased estimator of CA is given by

CCVA = CA + (CG − CG)

which becomes the point estimate to be used in Step 4 of section C.1. Using CG as acontrol variate reduces variance because it ‘‘steers’’ the estimate toward the correctvalue.


COMPARISON

The spreadsheet file AsianCallVarReduction.xls contains a Crystal Ball model shownin Figure C.1 for estimating the value of an average-price Asian Call option, alongwith reports on the performance of the model for options having different strikeprices and volatilities of the underlying asset. Figure C.2 shows the distribution of thediscounted cash flows for simulations with no variance reduction, antithetic variates(AV), and control variates (CV) for an at-the-money Asian Call option havinginitial price = $55, strike price = $55, underlying volatility = 30%, and time toexpiration = 126 days. All of these distributions have means that are estimates ofthe option’s value, but only the distribution of values from the simulation with novariance reduction resembles the distribution of cash flows that might be generatedby the option. The other three simulations yield distributions that have the desiredmean, but the shapes of the distributions are not indicative of the potential cashflows of the option. However, the mean standard errors computed from each of thefour distributions are comparable.

Table C.1 shows the mean, standard error, and percentage of reduction instandard error for each estimation method when the simulation model was run forN = 1, 000 trials. The antithetic variates nearly doubled the precision (halved thestandard error). The control variates method gave an estimate that is roughly twentytimes more precise than (has a standard error error that is 4.8% of) that achievedwith no variance reduction.

FIGURE C.1 Crystal Ball model for estimating the value of an average-price Asian Call option.

Variance Reduction Techniques 247

FIGURE C.2 Distribution of the discounted cashflows for simulations with no variance reduction,antithetic variates (AV), and control variates (CV) foran at-the-money Asian call option having initialPrice = $55, strike Price = $55, underlyingvolatility = 30 percent, and time to expiration = 126days.


TABLE C.1 Means, standard errors, and increases in precision from thesimulation model in AsianCallVarReduction.xls using no variancereduction, antithetic variates (AV), control variates (CV) to obtain estimatedvalues of an average price Asian call option having initial price = $55, strikeprice = $55, underlying volatility = 30%, and time to expiration = 126days.

StandardEstimation Method Mean Precision Error Increase

No Variance Reduction 3.3611 0.0476 1.00Antithetic Variates (AV) 3.3580 0.0235 2.03Control Variates (CV) 3.3102 0.0022 21.64

CONCLUSION

Variance reduction techniques offer potentially large increases in the precision ofestimated derivative values. The method of antithetic variates (AV) is generally lesseffective than control variates (CV), but AV can be easily applied to more types ofderivatives than CV because CV requires that a control variate is available, such asthe value of the geometric-average option that was used here.

Interest in use of Monte Carlo methods for derivatives pricing is increasingbecause of the flexibility of the method in handling complex financial instruments.Monte Carlo simulation will continue to gain appeal as financial instruments becomemore complex, workstations become faster, and simulation software is adopted bymore users. The use of variance reduction techniques along with the greater powerof today’s workstations can help to reduce the execution time required for achievingacceptable precision to the point that simulation can be used by financial traders tovalue derivatives in real time.

APPENDIX DAbout the Download

INTRODUCTION

This appendix provides you with information on the contents of the download thataccompanies this book.

ACCESSING THE DOWNLOAD

1. Visit http://textbook.crystalball.com2. Enter the code given on the card at the back of this book

SYSTEM REQUIREMENTS

Please refer to the Decisioneering web site, www.crystalball.com, for the latestsystem requirements.

WHAT’S INCLUDED IN THE DOWNLOAD

The following sections provide a summary of the software and other materialsincluded in the download

Excel Worksheets

Any Excel worksheets from the book are in the folder named ‘‘Content.’’ Theseworksheets are provided for your reference so that you may track the course ofthe book using Microsoft Excel and build financial models of your own using theseworksheets as templates. Note that each file, however, typically contains additionalinformation in different worksheets within that file.

Note: Many popular spreadsheet programs are capable of reading MicrosoftExcel files. However, users should be aware that a slight amount of formatting mightbe lost when using a program other than Microsoft Excel. Also, the Crystal Balladd-in works only with Microsoft Excel.

249


Crystal Ball Trial

Crystal Ball software transforms Microsoft Excel spreadsheets into dynamicmodels that solve almost any problem involving uncertainty, variability and risk.Included with this book is a time-sensitive license of Crystal Ball Professional thatwill expire 180 days from the date of installation. Crystal Ball Professional includesMonte Carlo simulation plus advanced capabilities of OptQuest, to search foroptimal solutions, CB PredictorTM to create accurate predictive models and ExtremeSpeed, to run simulations up to 100 times faster.

To access the software, follow the download instructions on the insertedcard. If you experience difficulties downloading or installing the software, pleasecontact Crystal Ball’s Technical Support Dept. at [email protected] orhttp://support.crystalball.com. Technical support is available only through e-mailand the Decisioneering web site. Crystal Ball and Decisioneering are registeredtrademarks of Decisioneering, Inc.

CUSTOMER CARE

Technical support is available only through email and the Decisioneering web site.For technical support, contact us at [email protected] or http://support.crystalball.com.

Glossary

arbitrage The purchase of securities on one market for immediate resale on another in orderto profit from a price discrepancy.

assumption An estimated value or input to a spreadsheet model.assumption cell A value cell in a spreadsheet model that has been defined as a probability

distribution using Crystal Ball’s Distribution Gallery.CDF Cumulative distribution function, which gives the probability that a variable will fall

at or below a given value.certainty bands In a trend chart, a graphic depiction of a particular certainty range for each

forecast.certainty level The percentage of values in the certainty range compared to the number of

values in the entire output distribution.certainty range The linear distance for the set of values between the certainty grabbers on

the forecast chart.coefficient of variability also coefficient of variance or coefficient of variation A measure

of relative variation that relates the standard deviation to the mean. Results can berepresented in percentages for comparison purposes.

continuous probability distribution A probability distribution that describes a set of unin-terrupted values over a range. In contrast to the Discrete distribution, the Continuousdistribution assumes there are an infinite number of possible values.

correlation In Crystal Ball, a dependency that exists between assumption cells.correlation coefficient A number between −1 and 1 that specifies mathematically the degree

of positive or negative correlation between assumption cells. A correlation of 1 indicatesa perfect positive correlation, −1 indicates a perfect negative correlation, and 0 indicatesthere is no correlation.

cumulative frequency distribution A chart that shows the number or proportion (or per-centage) of values less than or equal to a given amount.

decision variable A Crystal Ball variable in your model that you can control.derivative security A financial instrument whose price is derived from that of another

financial security.deterministic model Another name for a spreadsheet model which yields single-valued

results.discrete probability distribution A probability distribution that describes distinct values,

usually integers, with no intermediate values. In contrast, the Continuous distributionassumes there are an infinite number of possible values.

display range The linear distance for the set of values displayed on the forecast chart.dominant A relationship between distributions in which one distribution’s values for all

percentile levels are higher than another’s. (see also Subordinate)entire range The linear distance from the minimum forecast value to the maximum forecast

value.

251


forecast A statistical summary of the assumptions in a spreadsheet model, output graphicallyor numerically.

forecast cell A formula cell that has been defined as a forecast and refers either directly orindirectly to assumption cells.

forecast definition The forecast name and parameters assigned to a cell in a Crystal Balldialog.

forecast formula A formula that has been defined as a forecast cell.forecast value also trial A value calculated by the forecast formula during an iteration. These

values are kept in a list for each forecast, and are summarized graphically in the forecastchart and numerically in the descriptive statistics.

formula cell A cell that contains a mathematical formula.frequency also frequency count The number of times a value recurs in a group interval.frequency distribution A chart that graphically summarizes a list of values by subdividing

them into groups and displaying their frequency counts.goodness-of-fit A set of mathematical tests performed to find the best fit between a standard

probability distribution and a data set.grabber also certainty grabber and truncation grabber A control that lets you use the mouse

to change values and settings.group interval A subrange of a distribution that allows similar values to be grouped together

and given a frequency count.iteration also trial A three-step process in which Crystal Ball generates random numbers

for assumption cells, recalculates the spreadsheet model(s), and displays the results in aForecast Chart.

kurtosis The measure of the degree of peakedness of a curve. A normal distribution curvehas a kurtosis of 3.

Latin hypercube sampling In Crystal Ball, a sampling method that divides an assumption’sprobability distribution into intervals of equal probability. The number of intervalscorresponds to the Minimum Sample Size option available in the Run Preferences dialog.A random number is then generated for each interval. Compared with conventionalMonte Carlo sampling, Latin hypercube sampling is more precise because the entirerange of the distribution is sampled in a more even, consistent manner. The increasedaccuracy of this method comes at the expense of added memory requirements to hold thefull Latin hypercube sample for each assumption.

mean The familiar arithmetic average of a set of numerical observations: the sum of theobservations divided by the number of observations.

mean standard error The Standard Deviation of the distribution of possible sample means.This statistic gives one indication of how accurate the simulation is.

median The value midway (in terms of order) between the smallest possible value and thelargest possible value.

mode That value which, if it exists, occurs most often in a data set. In a continuousprobability distribution, the mode is the number on the horizontal axis lying beneaththe highest point on the pdf curve. In a discrete probability distribution, the mode is thevalue having the greatest probability of occurrence.

model sensitivity The overall effect that a change in an assumption cell produces in a forecastcell. This effect is solely determined by the formulas in the spreadsheet model.

Monte Carlo simulation A system which uses random numbers to measure the effects ofuncertainty in a spreadsheet model.

Glossary 253

PDF Probability density function that represents the probability that an infinitely smallvariable interval will fall at a given value.

probabilistic model A system whose output is a distribution of possible values. In CrystalBall, this system includes a spreadsheet model (containing mathematical relationships),probability distributions, and a mechanism for determining the combined effect of theprobability distributions on the model’s output (Monte Carlo Simulation).

probability (Classical Theory) The likelihood of an event.Probability Distribution also Distribution A set of all possible events and their associated

probabilities.rainbow option An option whose value depends on more than one source of uncertainty.random number A mathematically selected value which is generated (by a formula or

selected from a table) to conform to a probability distribution.random number generator A method implemented in a computer program that is capable

of producing a series of independent, random numbers.range The difference between the largest and smallest values in a data set.rank correlation also Spearman’s rank correlation A method whereby Crystal Ball replaces

assumption values with their ranking from lowest value to highest value using the integers1 to N prior to computing the correlation coefficient. This method allows the distributiontypes to be ignored when correlating assumptions.

real option An option involving a real asset.relative probability also relative frequency A value, not necessarily between 0 and 1, that

indicates probability when used in a proportion.reverse cumulative frequency distribution A chart that shows the number or proportion (or

percentage) of values greater than or equal to a given amount.risk The uncertainty or variability in the outcome of some event or decision.seed value The first number in a sequence of random numbers. A given seed value produces

the same sequence of random numbers every time you run a simulation.sensitivity The amount of uncertainty in a forecast cell that is a result of both the uncertainty

(probability distribution) and model sensitivity of an assumption cell.sensitivity analysis The computation of a forecast cell’s sensitivity with respect to the

assumption cells.skewed An asymmetrical distribution.skewed, negatively A distribution in which most of the values occur at the upper end of the

range.skewed, positively A distribution in which most of the values occur at the lower end of the

range.skewness The measure of the degree of deviation of a curve from the norm of a asymmetric

distribution. The greater the degree of skewness, the more points of the curve lie toeither side of the peak of the curve. A normal distribution curve, having no skewness, issymmetrical.

spreadsheet model Any spreadsheet that represents an actual or hypothetical system or setof relationships.

standard deviation The square root of the variance for a distribution. A measurement of thevariability of a distribution, i.e., the dispersion of values around the mean.

strike price The contractually agreed amount for which the underlying asset may be tradedin an option contract.

subordinate A relationship between distributions in which one distribution’s values for allpercentile levels are lower than another’s. (see also Dominant)


trial also iteration A three-step process in which Crystal Ball generates random numbersfor assumption cells, recalculates the spreadsheet model(s), and displays the results in aForecast Chart.

trial as used to describe a parameter in certain probability distributions. The number oftimes a given experiment is repeated.

truncation The specification of an upper limit, a lower limit, or both on the range of valuesto be generated from a Crystal Ball assumption.

value cell A cell that contains a simple numeric value.variable A quantity that can assume any one of a set of values and is usually referenced by

a formula.variance The square of the standard deviation; i.e., the average of the squares of the

deviations of a number of observations from their mean value. Variance can also bedefined as a measure of the dispersion, or spread, of a set of values about a mean. Whenvalues are close to the mean, the variance is small. When values are widely scatteredabout the mean, the variance is larger.

virtual memory Memory which uses your hard drive space to store information after yourun out of random access memory. Virtual memory supplements your random accessmemory.

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Index

AAllen, F., 111Amram, M., 189Antikarov, V., 189Antithetic variates, 242–247Arbitrage, 170–171Arnold, B. C., 224Artzner, P., 142Asipu people, 2Assumptions, Crystal Ball, 29

Bernoulli, 36–39, 202, 205, 213, 214,233

Beta, 5, 203Beta binomial, 206Binomial, 39–40, 203, 204, 233Cauchy, 228Chi-square, 213Custom, 206Discrete uniform, 40–43, 208Double Exponential, 211, 220Doubly Exponential, 220Erlang, 213Exponential, 210, 214Gamma, 211, 212Geometric, 203, 213, 214Hypergeometric, 214Laplace, 211, 220Logistic, 216Lognormal, 5, 52–53, 217Maximum Extreme, 219Minimum Extreme, 220Negative binomial, 203, 214, 221Normal, 47–51, 206, 216, 217, 223Normal, mixtures of, 50–51Pareto, 224

Poisson, 5, 225Rayleigh, 232Sech-squared, 217Student’s t, 226Triangular, 30, 45–47, 228Uniform, 43–45, 229, 234Weibull, 211, 230Yes-No, 36–39, 122, 202, 210, 232

Autocorrelation, 149–153Autoregressive model, 163–168Avramidis, A., 176Ayyangar, A. A. K., 229

BBachelier, L., 159Baker, K. R., 194Balakrishnan, N., 201, 216Balance sheet, 125Batch fit, 68–70, 155, 157Batch fit tool, 136Bell, S., 228Benefits of Crystal Ball, 9Beta function, 204Black, F., 169, 171, 188Black-Scholes model, 170–171Blattberg, R. C., 121, 124Bodie, Z., 132Bortkiewicz, L. von, 226Bowman, E. H., 189Boyle, P. P., 243Brabazon, T., 189Bratley, P., 234Brealey, R. A., 111Broadie, M., 176, 243

263

264 INDEX

Brown, L., 2Brown, R., 159Brownian motion, 159

CCapital budgeting, 111–121

Risk analysis in, 116–121Center for Research in Security Prices,

132Chart windows, 102Coefficient of variability, 19Common size statement, 125Conditional Tail Expectation, 142Control variates, 242–247Cooper, R. G., 199Copeland, T., 189Copeland, T. E., 187, 189Correlation tool, 117Correlation, Pearson, 63–64Correlation, Spearman, 64, 65Correlation, specifying, 63–67Covello, V. T., 2Credibility, 35Cumulative distribution function,

201Customer lifetime value, 121–124

DData, use of historical, 54–61Decision table, 73–89

One decision variable, 73–79Two decision variables, 79–89

Decision table tool, 21Decision variables, defining, 71–73Define Assumption, 30Define Forecast, 32Delbaen, F., 142Distribution gallery, 30Distributions, fitting to data, 57–58Dixit, A. K., 187

Duffie, D., 159, 171Dwyer, R. F., 121

EEber, J. M., 142Edgett, S. J., 199Efficient markets hypothesis, 146Einstein, A., 159Embrechts, P., 152, 168Error

Model, 34Nonsampling, 34Sampling, 34Simulation, 35Sources of, 33

Esrey, W. T., 180Evans, M., 201Excel functions

CB.Beta2, 204CB.Beta, 204CB.Binomial, 205CB.Custom, 206CB.DiscreteUniform, 210CB.Exponential, 211CB.Gamma, 213CB.Geometric, 213CB.GetForePercentFN, 143CB.GetForeStatFn, 14CB.Hypergeometric2, 216CB.Logistic, 216CB.Lognormal2, 218CB.Lognormal, 218CB.MaxExtreme, 220CB.MinExtreme, 221CB.NegBinomial, 222CB.Normal, 224CB.Pareto, 225CB.Poisson, 226CB.StudentT, 227CB.Triangular, 229CB.Uniform, 230

Index 265

CB.Weibull, 232CB.YesNo, 233IF, 179IRR, 107NORMDIST, 4, 50NPV, 106ROUNDDOWN, 206SUMPRODUCT, 124VLOOKUP, 55

Expected Tail Loss, 142Exponentiated Brownian motion,

159

FFan, J. Q., 168Financial modeling, defined, 2Fishman, G. S., 234, 239, 243Ford, G. R., 158Forecast, Crystal Ball, 29

Filter values, 143Fox, B. L., 234Frequency chart, 12

Cumulative, 13Reverse cumulative, 14

Frey, R., 152, 168Futures contract, 187

GGamma function, 204Gauss, C. F., 146Geometric Brownian motion, 159–162,

171, 185Estimating parameters of, 160Generating correlated stock prices with,

162Generating stock prices with, 159

Geske, R., 176Getz, G., 121, 124Glasserman, P., 176, 234, 243Goedhart, M., 2

Goodness-of-fit testing, 58–61Eyeball test, 60–61

Gosset, W. S., 6Gupta, A. K., 203

HHaenlein, M., 124Hardy, M., 142Hastings, N., 201Heath, D., 142Hertz, D. B., 110Historical data unavailable, 62–63Hot keys, 14Hull, J. C., 169, 171, 176Hyden, P., 176

IIbbotson Associates, 132, 157Income statement, 125Internal rate of return, 105–124

JJensen’s Inequality, 121Jiang, R., 230Johnson, H. E., 176Johnson, N. L., 201

KKane, A., 132Kaplan, A. M., 124Keenan, P. T., 187Kelton, W. D., 201, 234, 239,

243Kemna, A. G. Z., 244Kemp, A. W., 201Kester, W. C., 188Keyboard shortcuts, 14Klein, E., 3

266 INDEX

Kleinschmidt, E. J., 199Knuth, D. E., 237Koller, T., 2Kotz, S., 201Kulatilaka, N., 189Kurtosis, 18

Excess, 224

LLaPlace, S., 3Latin hypercube sampling, 101, 240Law, A. M., 201, 234, 239, 243Leslie, K. J., 191Limitations of Crystal Ball, 9Longstaff, F. A., 176

MMacrae, N., 8Marcus, A. J., 132Marmorstein, H., 183Maximum, 20McDonald, R. L., 169, 171, 181, 187McNeil, A. J., 152, 168Mean, 15Mean Excess Loss, 142Mean Shortfall, 142Mean standard error, 20, 242Mean-reverting model, 163–168Median, 16Melicher, R. W., 105Merton, R., 169Meyers, S. C., 111Michaels, M. P., 191Miller, L., 189Min, X., 230Minimum, 19Mode, 17Model Risk, 34Modulo reduction, 234Mortality table, 2001 CSO, 137Moskowitz, G. T., 189Mumpower, J., 2

Mun, J., 189, 190Myers, S. C., 188

NNadarajah, S., 203Net present value, 105–124Norton, E. A., 105Number of trials to run, 96

OOppenheim, L., 2Option

American, 170, 176Asian, 180–182, 243–247Barrier, 179Bermudan, 176–178Bull spread, 182Call, 170–172Digital, 178European, 170Exotic, 170, 178–182Put, 170–172, 187

OptQuest, 89–94Constraint, 89Constraints window, 91Forecast selection window, 91Forecast statistic, 90Objective, 90Options window, 92Performance graph, 92Requirement, 90Status and solutions window, 92

Ore, O., 3Overlay chart, 25, 149

PPareto, V., 224Park, C. S., 189Pascal, B., 3Patel, J. K., 223Peacock, B., 201

Index 267

Percentiles view, 20Pindyck, R. S., 187Pitman, J., 201Portfolio insurance, 173Portfolio models, 132–139Powell, S. G., 194Prabhakar Murthy, D. N., 230Priestley, M. B., 150Principal-protected instrument, 183–185Probability density function, 201Probability distributions

See Assumptions, Crystal Ball, 201Proctor, K. S., 2Pseudorandom numbers, 236

QQuine, M. P., 226

RRandom number generation, 98, 234

Crystal Ball, 237Excel, 237Linear congruential generators, 235–237Specifying seed, 237

Random variate generation, 234, 238–241Inverse-transform method, 238–239Latin hypercube sampling, 240–241

Random walk process, 148–149Additive, with drift, 153–156Multiplicative, 156–158Vector, 154–156

Range, 20Read, C. B., 223Real options, 186–200

Analysis, 186–192Analysis toolkit, 190Categories of, 187Financial versus, 187Insights from Black-Scholes model, 190

Real options valuation tool, 192–200Steps in using, 192–195Use in new product development,

198–199

Value added by using, 195–198Reichheld, F. F., 121Retirement planning, 135–139Risk management, 8Risk-neutral pricing, 170Robinson, T., 183Rubinstein, M., 159Rubinstein, R. Y., 8Run mode, 101Run preferences

Options, 103–104Sampling, 98–101Speed, 101–102Statistics, 104Trials, 95–98

SSampling from a distribution, 56Sampling method, 101Sampling, direct, 54–56Samuelson, P., 159Schoder, D., 124Scholes, M., 169, 171, 188Schrage, L. E., 234Schulte, D., 183Schwartz, E. S., 176Seneta, E., 226Sengupta, C., 2, 125–131Sensitivity analysis, 116, 126Sensitivity chart, 129Serial correlation, 150Simulation cycle, 95Simulation modeling process, 28Simulation study, 33Simulation study versus statistical study, 34Single step, 32Skewness, 17Solver, Excel’s, 134Spider chart, 114–116Spreadsheet models

AKGolf.xls, 28–33Accumulate.xls, 11–26, 98Analytic.xls, 4

268 INDEX

Spreadsheet models (continued)AppendixA.xls, 206AsianCall.xls, 181AsianCallVarReduction.xls, 243,

245–247AssetOrNothingCall.xls, 179BatchFit.xls, 68–70BermuPut.xls, 176Bernoulli.xls, 36Binomial.xls, 41BullSpread.xls, 182, 183CLT.xls, 48CVaR.xls, 143–144CVaRSubadditivity.xls, 145CorrelatedGBM.xls, 162DirectSampling.xls, 54ETFs.xls, 158EsreyOptions.xls, 180EuroCall.xls, 172EuroPut.xls, 172FiveTosses.xls, 37InverseTransform.xls, 239Kurtosis.xls, 18LCGExamples.xls, 235LifetimeValueModel.xls, 122Lognormal.xls, 53, 57, 58, 60MixtureModel.xls, 51NPV.xls, 106NPVModels.xls, 108Newton.xls, 41–43Normal.xls, 51OneYearTreasuryYields.xls, 167PearsonSpearman.xls, 64, 66Portfolio.xls, 132PortfolioVaR.xls, 141PrincipalProtectedInstrument.xls,

184Profit.xls, 5Project.xls, 79–94, 119RandomWalk.xls, 146–149, 151RandomWalkWithDrift.xls, 154–156SPY.xls, 157SPYwithGBM.xls, 161, 162ScooterNPV.xls, 111

ScooterSimulation.xls, 117, 120Sengupta.xls, 126, 127Sengupta2.xls, 128, 129Sengupta3.xls, 130TenYearTreasuryYields2005.xls, 164Triangular.xls, 45TwoCorrelatedAssets.xls, 71, 72Uniform.xls, 44VFH.xls, 174

Standard deviation, 17Statistical study, 33Statistics

Coefficient of variability, 19Kurtosis, 18Maximum, 20Mean, 15Mean standard error, 20, 242Median, 16Minimum, 19Mode, 17Range, 20Skewness, 17Standard deviation, 17Trials, 15Variance, 17

Statistics view, 14Stop on calculation errors, 96Stop when precision control limits are

reached, 97Strike price, 169

TTail VaR, 142Tasche, D., 145Thomas, J. S., 121, 124Threshold values, 84Tornado chart, 112–116, 126,

194Trend chart, 23Trent, W., 183Trials, 15Trigeorgis, L., 187Truncation, 202Tsay, R. S., 150, 167, 168

Index 269

UUlam, S., 8Uryasev, S., 142

VValidation, 35Value at Risk, 140–145Value at Risk, Conditional, 142–145Variance, 17Variance reduction, 242–247

Antithetic variates, 244Control variates, 244

Verification, 35

von Neumann, J., 8Vorst, A. C. F., 244

WWessels, D., 2White noise process, 146–148

Checking for, 152Willmott, P., 159, 162Wilmott, P., 169, 171, 187

YYao, Q. W., 168

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